Statistics for Data Science_II Live Stream

>> MA 1004: Hello everyone. Let's wait for 5 minutes and We'll start the session, okay? >> 24F2005774 ABHISHEK KUMAR: Okay. >> MA 1004: Hello. Let's start the session. >> SHAHRUKH HASHMI: That. Yes, ma'am. >> MA 1004: Yeah. so in week three, are we are going to study about the expectation variants standard deviation, correlation covariance, and lastly markers inequality. and turbitions inequality Ity. So first comes the expectation. Okay. so what is expectations if we have to summarize the data, okay? If we have a lot set of data then we have to summarize it, right? So we have to summarize it in one number then we calculate the expectation. So how do we calculate the expectation expectation? It is denoted by E of X. Which is equal to submission of. T. F x. T. Okay, so where This T. Is the variable you can see and f x the function of that variable. right? Is it fine? >> ATUL SAWARN: Yes.

>> MA 1004: Okay. so the expectation of X is also known as the mean of X, or you can say, if you have to calculate the average value of things, Okay. and it has the same unit as X, right? Okay. So, for example, if you have to calculate the expectation of a BUN, knowledge, Okay. but knowledge, the p as a parameter. So in but only trial what? What are the probabilities? >> Pyari Singh K: P, and 1 minus p. >> 24F2006261 SHIVAM KUMAR: Two brothers. >> MA 1004: Yeah, E. And 1 minus t. Okay. So if the probabilities Med one for p and 1 minus P is 0. >> 24F2006261 SHIVAM KUMAR: 0. >> MA 1004: Right. So if you have to calculate the expectation, it will be the value that is 1 into probability is p. Right. Plus Zero. multiply its probabilities, 1, minus P. So it will be equal to Is it clear? Yeah. so if you have to now other example, if I have to calculate the expectation of if if I'm throwing a die.

Okay? If I'm rolling a diet, then what is the expectation, right? So what are the chances? The chances are one, two, three, four, five. and six. Right. And the probabilities are for one. The probability is >> Pyari Singh K: One by six. >> MA 1004: One by six. Similarly, for all of them, the probability will be 1 by 6 correct. So, if I have to calculate the expectation, that is the average value, it will be One. >> Pyari Singh K: One into a 6 plus 2 into 1 by 6. >> MA 1004: Yeah, one by six into one plus two into one by six plus three into one by six. Similarly it goes to 6 into 1 vices, then this total will be equal to 3.5. so the probability is three. Sorry the expectation is 3.5 Right. So this is how you calculate the expectation.

Okay. so there are many, you can say Like there are many distributions, Okay. So for like geometric distribution for Porsche distribution. For Bernoulli. it was the expectation was Right. >> Pyari Singh K: Np. >> MA 1004: But not only. >> Pyari Singh K: Yeah. >> MA 1004: Yeah. >> Pyari Singh K: yeah. P. >> MA 1004: Np. okay, for geometric, it will be one Bible. Right. And for sure, it will be. >> Pyari Singh K: One lambda. >> MA 1004: Laughter. Okay. so these are the expectations. Okay. Now from the graph if we have to calculate, if you see the graph here, if you can see the graph, it's the Bernoulli. but knowledge distribution. and the p value is 0.3. Okay. so if we draw graph here, the expected value will be the. You can say, the point, the highest point you Right. This bar is having the highest from point, right? Compared to this. Correct. So this will be the expected value, which is 0.3. Okay. Now, but binomial is like we have binomial distribution.

It is given n and correct. So if we draw graph here, okay. so we see that expectation is six. That is expectation. Of binomial is n p, right? N is 20 into 0.3, which is 6. So the value, the expected value is 6, which if you can see, it is the higher point. the highest point you can see. So the expectation value or the average value is the highest point in the graph Fine. Is it clear? Now, let's come. ah, there are some properties of expectation. Okay. So the first property is The first properties. the expectation of any constant value is always Second. The X, like, if we having a The X. It is negative value. It is taking negative 1, right? then the expectation is Greater than 0. No, sorry. Sorry. >> PRAKAASH A S: Greater than normal. >> MA 1004: No, no. If the probability of x greater than zero is one.

Then the expectation or is greater than that 0. That that is the expected. The x values, always taking non-negative values. it is not taking any negative values. okay, so if it is taking only negative non-negative values and the expectation will also be non-negative. Right. >> P Baskaran Nadar: Like, sorry. >> MA 1004: now, there comes with Theorem >> P Baskaran Nadar: Why is that? So >> MA 1004: Why is that? So >> P Baskaran Nadar: Probability is always greater than zero. >> MA 1004: like, if you can, The probability that the X will that excess, taking always positive value is one, okay? So the expectation will also be positive right? Yeah. >> P Baskaran Nadar: Okay. >> MA 1004: like in the above example, if we have seen throwing a dice the all the values of x are positive, right? And the probabilities is positive. >> P Baskaran Nadar: What you're saying is X doesn't have a negative value at all. >> MA 1004: yeah, if X is not having negative values, >> PRAKAASH A S: Then next is not having.

>> MA 1004: Yeah. >> 24F2006261 SHIVAM KUMAR: Rotation value is more memory. >> PRAKAASH A S: Explorer also, >> MA 1004: Sorry. >> 24F2006261 SHIVAM KUMAR: ah, >> PRAKAASH A S: the next non-negative, then only >> MA 1004: Yeah. >> PRAKAASH A S: expectation is not negative. >> MA 1004: greater than >> 24F2006261 SHIVAM KUMAR: what I was saying is now, this is expectation. value, is it is an average man. So, average is always positive. >> MA 1004: Yeah. Positive one. Yeah. It can't be negative. >> 24F2004702 RAVI SHEKHAR: Mum. I have a doubt in that. there are few questions like, for example, given a probable probability of. I mean if I toss a coin and I am getting a probability of head as half and let's say, I toss a coin and when the head comes I get ah I get five rupees for getting head and I get mine.

Minus 10 rupees for tails, in that case. My, my average would always be can be And let's say that. >> P Baskaran Nadar: She put a condition X doesn't become negative in this. >> 24F2004702 RAVI SHEKHAR: Someone then the statement becomes wrong. >> MA 1004: No. >> 24F2004702 RAVI SHEKHAR: now, he average cannot ne >> P Baskaran Nadar: That this this is not applicable in that case because x is negative in your case. >> MA 1004: Yeah. here I'm talking about the X is only taking positive values. That's it. That's not taking any negative. values. >> 24F2004702 RAVI SHEKHAR: Okay. >> MA 1004: Fine.

>> 24F2004702 RAVI SHEKHAR: Yes, ma'am. >> MA 1004: Okay. So yeah. And the third theorem, it says that if you are having a suppose, if you are having x 1 x 2 and xm we are having a joint. They are having a joint PMF. As f x 1. This is a joint pm of X1 x2 and x. and random variables. Okay. Then let y be a function of x 1 comma x2 xm, okay with Green. just t by and pmfs fy. So if we have to calculate the expectation of y that is we have to calculate the expectation of function of x 1 comma x 2 xl. Okay. Then it will be equal to summation of This function that is x 1 x n. multiplied by the Joint PMM of PMF of X 1 to x.

Got it. Everybody got it. This theorem Right. >> RAM KRISHNA SINGH: Yeah. >> MA 1004: means like if we have to calculate a, the expectation of why, which is a function of x, okay? Then we can calculate the we don't have to calculate the joint payment. We can simply calculate by the that functionally. Okay, let's for example, ah, we can say the values that x is taking Minus 2 comma minus 1, 0 1 and 2.

Okay. the probability is for minus 1. Minus 2 is 1 by 5. Minus 1 is 1 by 5 for 0, is 1 by 5. Similarly, for the other two's, okay? Now, if we have to calculate that is the Y is, X Square, right? So if you have to calculate the expectation of y, then it will be equal to That is the function, right? The function is x square. So x square is that is minus 2 square into the probabilities, 1 by 5. Correct. Plus that is minus 1 square into probability. That is 1 by 5. Similarly, we can calculate the other values. Square into 1 by 5 plus 2, square into 1 by 2. Is it clear? Here we are not calculating the joint.

We have. That is we are not calculating the X square first, right? So if it's x squared, then the values, the square will take is 0 comma, 1 comma 4. Right. Yeah. >> Pyari Singh K: Yes. >> MA 1004: so the probability of 0 will be 1 by 5 only for one. It will be. Two by five. and for 4. >> RAM KRISHNA SINGH: To it. Thank you. >> MA 1004: 2.5. >> 24F2006261 SHIVAM KUMAR: 2 times in repeating the values. >> MA 1004: Yeah. >> RAM KRISHNA SINGH: What? I will. >> MA 1004: Okay. so here we are not like we are not putting the joint pmf. We are just putting the values of the function of x. The >> RAM KRISHNA SINGH: Okay. >> MA 1004: Right. so it is not necess that you have to calculate the joint PMF. Then you can like you have to calculate the expectation. Just putting the values of the function of x. you can calculate the expect. >> Pyari Singh K: Mom, in this case, if Gene fixes something like 3x, >> MA 1004: Ation. >> Pyari Singh K: What will be? how will we find expectation of Y? if the function is instead of x square, if it is something like 3x, >> MA 1004: See.

>> Pyari Singh K: So do we multiply to? Yeah. >> MA 1004: Yeah, there is one more property if you can click, if there is a one more property, expectation of C into x. That is A Constant is multiplied by some variable then it will be equal to c into expectation of this. >> Pyari Singh K: Oh yeah. so in that case, we can do this. So what if it is like ED cases like XY. but, >> MA 1004: Xy. like there are two random videos like, >> Pyari Singh K: Yeah, I am. >> MA 1004: X and Y. >> Pyari Singh K: we use this theorem there as well? >> MA 1004: Mmm. Yeah. I will get to that point. Okay? >> Pyari Singh K: Okay. >> MA 1004: So this clear? This Theorem third, theorem >> Pyari Singh K: Yes.

>> MA 1004: It example. >> Pyari Singh K: examples. That definition is kind. Yeah. >> MA 1004: Definitions confusing. >> Pyari Singh K: yeah, for me, it is >> MA 1004: Okay, what's the confusion? >> Pyari Singh K: He was trying to actually map it to other functions like 3x. So you gave you gave an example of you know, a different property.

Altogether. Can we use the same function? There is what I was. I mean, same, theorem, there is what I was thinking. >> MA 1004: Three X, right. >> Pyari Singh K: So that is also different function. >> MA 1004: Thank you. >> Pyari Singh K: right? >> MA 1004: Yeah, it's a different function. so like you can do it right.

Like three in two minus 2 into 1 by 5 plus 3. Into minus 1. So it will be only three into expectation of X right? See. >> Pyari Singh K: Okay, so if every if every around, every possible value, we change it into that function and then multiply >> MA 1004: Yeah this yeah. This multiply with my probability, plus 3 into minus 1. This is one function into 1 by 5. Similarly for the other cases >> Pyari Singh K: Okay. >> MA 1004: so, here like if you can see here, the three is common. So we are just taking out the common and Writing the whole part again. that's it. >> Pyari Singh K: Right.

>> MA 1004: And this is simply the expectation of X. >> Pyari Singh K: Correct. >> MA 1004: Getting. >> Pyari Singh K: Yeah. >> MA 1004: Yeah, that's it. >> Pyari Singh K: Yes. Yes. >> MA 1004: Now, it's clear. >> Pyari Singh K: ma'am. >> MA 1004: Everybody. >> 24F2006261 SHIVAM KUMAR: you draw the graph of this y equals to x square.

How are you going to plot this expertise expectations? and views on you? >> MA 1004: Sorry, what? >> Neha Kumari IIT Madras: Can you? >> MA 1004: which, >> Neha Kumari IIT Madras: On the fourth one. The. >> MA 1004: This one. >> Neha Kumari IIT Madras: Calculation. Oh yeah. Just Yes. Mom got it. you >> MA 1004: So the third property is clear to everyone. >> 24F2006261 SHIVAM KUMAR: Yeah. >> MA 1004: Can we pro? >> 24F2006261 SHIVAM KUMAR: and then I was telling that initially you started and you display some of the graph of this expectation and that you sold some. it will look like this. In this case when we have a function like this y, equals to 4 x square, which is a parabola, >> MA 1004: Hmm. >> 24F2006261 SHIVAM KUMAR: How are you going to draw? just the expectations that use on? Such.

>> MA 1004: Of X Square. Right now, I'm not sure. But on Tuesday, I'll clear this down, right? >> 24F2006261 SHIVAM KUMAR: Okay. >> MA 1004: Yeah. Okay. so, Three properties are done. Now, the fourth property is the linearity of expected value, okay? It says that the expectation of Constant. multiplied by some variable, it will be constant. multiplied. >> 24F1002806 TOMENDRA KUMAR SAHU: He? >> MA 1004: Okay, this is the fourth property and the fifth property is the expectation.

If you have been given to weight random variables that is x and y. So the expectation of X plus y, Will be expectation of X plus expectation of Y. Got it. It is not necessary that it should be independent or dependent. It's just that that we have been given to random variables x and y and if we have to find the value of Please mute yourself. so yeah, if like it is given that only like, we have been given to random variables x and y and we have to calculate this expectation of some of these two random variables.

Then it will be equal to the expectation of x plus expectation of why. Got it. Yeah. so now other like if it's given a x plus b y then it will be equal to >> Pyari Singh K: Aim to expectation of X plus B into >> MA 1004: Hmm. >> Pyari Singh K: of Y, >> MA 1004: Yeah, correct. >> 23F3004210 CAESAR PARTHO KARMOKAR: Then can you show us an example of I mean, for these theorem the number five, >> MA 1004: Theorem, uh, five.

Okay. >> 23F3004210 CAESAR PARTHO KARMOKAR: Yeah. >> MA 1004: so, Let's take an example. It's comma. Y are taking values. These are the particles. One common zero. 0 1. Minus one. 1. Okay. These are the values that x and y are takes, right? now like, I want to the function, there is one other function of x and y, which is X square, plus, x y, plus y square. Okay. So, if we have to calculate the expectation of this value, right? That is X square, plus, x y, plus Y square. okay? So just suppose that this is one variable. this is one variable and this is one variable here we had two variable the sum of two variables. If you have three, some of three variables then it will be equal to expectation of x square plus expectation of x y, plus expectation of Y squared.

Yeah. Till here, it's the, it's okay. Yeah. >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes, ma'am. >> MA 1004: Okay. now, the value that we have to calculate the expectation of x square Okay, so the X is the excess taking values. Zero one. and minus 1, right? Or. Yeah. so, the probabilities will be So 0 is taking. 2 by 6. 1. 3 by 6 and minus 1. 1 by 6. Okay. Similarly, the Y is taking values 0 comma One comma minus 1, right? So the probabilities will be 0 will be 2 by 6. For 1 will be. >> 23F3004210 CAESAR PARTHO KARMOKAR: How are you calculating this 2 by 6 >> MA 1004: Okay. >> 23F3004210 CAESAR PARTHO KARMOKAR: 3 by 6? >> MA 1004: See See, this is the This is the data. So here's this is representing the first week. The first point is representing the x, and the second is representing the one. Yeah. Similarly, for the other, this is representing the x and this is representing the y. So if you see in this data that what are the values of x, Just give me the values of X. >> 23F3004210 CAESAR PARTHO KARMOKAR: 0 1, minus 1. >> MA 1004: 0 and 1 and minus 1, right? >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes.

>> MA 1004: And how many points are there. One two, three, four, five six, right? So how many times X is taking from the 6 out of six? how many times the value 0 is occurring? >> 23F3004210 CAESAR PARTHO KARMOKAR: Twice. >> MA 1004: Twice, right. So >> 23F3004210 CAESAR PARTHO KARMOKAR: Twice. >> MA 1004: 2 by 6 similarly how many times one is? >> 23F3004210 CAESAR PARTHO KARMOKAR: Thrice. >> MA 1004: Price. this and minus 1. Ones. >> 23F3004210 CAESAR PARTHO KARMOKAR: Only ones. >> MA 1004: Right. So this is how I calculate. >> 23F3004210 CAESAR PARTHO KARMOKAR: Okay, man. Clear >> MA 1004: Yeah. >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes. >> MA 1004: Yeah. So you have to calculate the values of x Square. How how will we calculate the value of x square? The x variable is taking this one. So, tell me. >> Pyari Singh K: 0 square into 2 by 6 plus. 1 square into 3 by 6, plus Minus 1 whole square into 1 by 6. >> MA 1004: So this would be 3 plus 6. Plus 1 by 6, which is Four buses. this year. Yeah. >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes, ma'am. >> MA 1004: Similarly, we can calculate all the values that is X square expectation of x Y and Y square.

Yeah. so the values see the values of expectation of X squares. 4 by 6. Expectation of x y is Minus 1 by 6. And y squares. So it put these values. that is 4 by 6 minus 1, by 6 plus 4 by 6. so, it will get 7 by 6 as the expectation. >> Pyari Singh K: Somehow. Yeah. >> MA 1004: Got it. >> Pyari Singh K: How did it calculate that? >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes. >> Pyari Singh K: expectation of XY. So fast. Did you have it written or Do we? >> MA 1004: Yeah. >> Pyari Singh K: Okay. >> MA 1004: Yeah. >> 23F3004210 CAESAR PARTHO KARMOKAR: Now, can you show the xy part? X. Y part. How is how your calculating? >> MA 1004: The XY part. >> 23F3004210 CAESAR PARTHO KARMOKAR: Yeah. How you are getting minus 1 by 6. For x y. >> MA 1004: just, like, if you see, >> Pyari Singh K: I think we have to list down the possible.

>> MA 1004: So you can write it as Expectation of X and expectation. Of fine. If you can, like if you get like, the expectation of xy can be written as expectation of X. And expectation of one So, if you calculate the expectation of X, then multiplied it by the expectation of why you will get the value, as the value of expectation of x. And Right. >> Pyari Singh K: But this is possible only when it's independent, right? Okay. >> MA 1004: Yeah, this is an independent case.

Okay. >> Pyari Singh K: Okay. >> MA 1004: So, yeah. so this is how you can get >> JB ANMOL: You can begin your X square as well. >> MA 1004: Sorry, what? >> JB ANMOL: Same can be done for X square as well. It can be written as expectation of x into expectation of X, >> MA 1004: Yeah. >> Pyari Singh K: Know that clearly independent clearly dependent. I think. >> MA 1004: Yeah, it's X squared, right? We are just. The x and y are independent. the two variables are in the, but that the x is not dependent on why. That's the thing. >> JB ANMOL: Okay. >> PRAKAASH A S: Probably Ahm. >> Rahul: Great find. >> PRAKAASH A S: ed probability for which variable given X or y.

>> MA 1004: Okay, just give it. Just a second. Okay. I said it wrong. Okay, it's not independent, because C, X and Y. Are the points, right? 0? comma zero, one comma zero zero comma 1, right? These are the points. coordinates, right? Yeah. so it can't be independent. I'm very sorry about that. It's not independent. How because the values of Y is dependent on x, right? So if you're having the coordinates right, they are dependent. If you get having one value of x we can calculate the other value of y. So it's dependent, right? If you are having some function and if you have to calculate the coordinates of file, like, if we are having some linear function, x plus 3 is equals to y. So like if you have to find the coordinates, how how we used to calculate the coordinates, like the value of x and calculating, the value y. Right. So it is dependent the coordinates. They are dependent. They are not independent. Okay. so how did we get? So this will not apply.

Okay. So we calculated expectation of x y as The first, what was the data like it >> Pyari Singh K: 0. >> MA 1004: was zero comma, zero, the first multiplied by the water ball will be the probability one by six, right? One times it is appearing. Correct. Is it clear? >> 23F3004210 CAESAR PARTHO KARMOKAR: just, >> MA 1004: Yeah. Similarly the next point was one comma zero. multiplied by its probability. Similarly Zero comma one into one by six. plus 1 comma. one multiplied by 1 by 6 plus Minus 1. Comma. multiplied by 1 by 6. Plus 1 comma, minus 1, multiplied by 1 lessons. So here we have multiplying, the values of x and y, right? So we will multiply these values. These coordinates the values of x and y, so it will be 0 multiplied by 0. 0 multiplied by 1 by 6. Similarly. >> PRAKAASH A S: Of 0. >> MA 1004: 1 into 0 is 0 0. >> PRAKAASH A S: 0. >> MA 1004: Into 1 by 6 plus. >> PRAKAASH A S: 1 Basis. >> MA 1004: 1 into 1 by 6. Minus 1 into 1 minus 1 into 1 by 6 plus minus 1 into 1 vices This part is clear.

Yeah. >> Pyari Singh K: Last one is one. >> MA 1004: Yeah. >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes, ma'am. >> MA 1004: it will be minus 1. 1 into minus 1. >> PRAKAASH A S: Answer management basis. >> MA 1004: Right. So this will be zero, zero zero, this will be 1 by 6, this will be minus 1 by 6 and this will be minus less On. So this will get canceled. So our if the expectation is minus 1 by 6, >> JB ANMOL: Can you explain X Square? and Y Square? Once again, please? >> MA 1004: Okay, so this part is clear. >> JB ANMOL: Yes. >> Pyari Singh K: A normal. >> MA 1004: Yeah. >> Pyari Singh K: How did? >> MA 1004: What's about? >> Pyari Singh K: I understood the first part. How he wrote? >> MA 1004: Hmm.

>> Pyari Singh K: That's the definition. How did we simplify this? How is that? 0 into 1 by 6 plus? >> MA 1004: See we are calculating. The we are multiplying. The values of x and y. This is clear. This part. We are calculating the multiplication of x and y value, right? So excess taking z. So here, the coordinator, the x value is taking 0 and the wire is taking zero. So if you multiply, we will get 0. >> Pyari Singh K: Okay, so it's x y, okay. >> MA 1004: Yeah, this is XY. This is also xy. That is the points are getting multiplied. Now, it's clear. >> Pyari Singh K: Yes, yes. >> MA 1004: 1 into 0 0, into 1. 1, into 1 like that. Fine. >> Pyari Singh K: Ma. >> MA 1004: Okay, so this part is clear. Everybody. Okay. Some. It's good, right? Who had a doubt in expert? >> JB ANMOL: Mummy. >> MA 1004: Yeah. What, what is it now? >> JB ANMOL: And for a like the way, how we calculated? X y, same can be done for x square as well. Like, those points will be zero zero and one one.

Is it? >> MA 1004: like, >> JB ANMOL: if x and where, >> MA 1004: that's the thing. Yeah, we are just only taking the values of x, only right? And we are just squaring it up. >> PRAKAASH A S: Write this Ambrose with them. There will understand, I think. >> MA 1004: Study, what? >> PRAKAASH A S: Sample space of X and Y. >> MA 1004: Yeah. Like, here we are taking only the values of x. We are not taking the values of y, so we are just squaring the values of x that is 0 square, 1 square. and minus see in the sample. In the data, the coordinates, the x values. What are the x values? 0, 1, 0 1, minus 1 and 1. That is the excess. Taking 0 value. What? Like here 0, 1 and minus one.

These are the values that x is taking right? >> JB ANMOL: Okay, got it. >> MA 1004: Yeah. So we are just squaring the values of x. You're not taking. we are not taking the coordinates. We are not taking the y variable here. We are taking only the X square. that's it. The XY. >> JB ANMOL: Okay, okay. >> MA 1004: Yeah. Is this example clear? By everyone. >> PRAKAASH A S: And can you show the problem? Question. >> MA 1004: this is, >> PRAKAASH A S: Where is your question? >> MA 1004: Yeah, this one. See if x and y are taking values like this. >> PRAKAASH A S: Oh, that that itself. Okay, the action we are okay. System. produce. >> MA 1004: Fine. >> JB ANMOL: Just the pmf, right? >> MA 1004: Yeah, this is the values that x and y are taking.

That's it. >> JB ANMOL: Okay. >> MA 1004: And see. Can I proceed for the? >> JB ANMOL: Yes. >> MA 1004: Okay. So like like, This property you if you applying by see in Bernoulli, the expectation was p, right? What is binomial Like if X is following Bernoulli, then what is binomial? That is x 1 comma. Extrude till x n. Right. It is formed following but binomial distribution. Correct? With n and p as the parameter. So what will be the expectation? >> PRAKAASH A S: Can be. >> MA 1004: How how did we calculate it? See in Bernoulli, the expectation of x was P. So in binomial, it will be some of all the trials that is x1 plus X2 plus till xn, right? >> Pyari Singh K: Yes.

>> MA 1004: Is it? Yeah. so, the like, We will follow the property. That is X1 plus sorry. so, the expectation will be expectation of x 1 plus expectation of x 2 till Expectation of xn and all these like x1 x2. They are calling biology with expectation P, right? So, the expectation of x 1 is >> 23F3004210 CAESAR PARTHO KARMOKAR: B. >> MA 1004: e, plus expectation of x2sp till This. So how many values are there? There are. N values. That is n p. This is how we calculate the expectation of binomial. With this property, fine. Okay. Now comes the zero. mean random variable. What is 0 mean? Random variable zero mean random variable is that in which like if y is a random variable then x, minus expectation of x is a 0 mean random variable like if a random variable x is subtracted by is expectation of like, if we calculate the expectation of y, then, So, the expectation is zero.

Here. Why is denoted as x minus expectation of x. See if y is a random variable like why is denoted as x minus expectation of x? And if you are going to calculate the expectation of y, then it should be equal to See, the value of the expectation of x. Minus expectation of x only right. So this will be 0 only.

You getting it. like, zero mean random Variable says, >> 24F2004702 RAVI SHEKHAR: Yes, ma'am. >> MA 1004: that if, if there is one random variable, Okay. With expectation of x as 0. So this is, this is known as zero mean randomly. That's it. if the expectation of any random variable is zero, that means it's a zero mean randomly. Right. So if we are like, if you are finding out the expectation of X minus expectation of X, Right, if you are just subtracting the random variable my with the expectations and the value is 0, Getting my point. >> Rahul: Number one that we expectation of expectation of x. when you break the second last. >> 24F2004702 RAVI SHEKHAR: But that is a constant now, expectation of a constant is always >> MA 1004: Yeah, this is a constant value. so, expectation of a constant value is, See only.

>> Rahul: The con. >> MA 1004: Yeah, that is expectation of x. So if >> Rahul: Okay. >> MA 1004: you're subtracting expectation of X from the expectation, it will be zero. Only >> Rahul: Okay. >> MA 1004: So till here, anybody having any doubt in any topic? >> PRAKAASH A S: The given data is expedition of x equal to 0. Expectation of 0 is 0. So that's what you want to write. >> MA 1004: Sorry, what? >> PRAKAASH A S: What is a given data x? and the expectation of x equal to 0. >> MA 1004: No, no. >> PRAKAASH A S: Is it? >> MA 1004: it's no zero mean random will say is that if there is one random variable x With. Expectation of X as 0, if that random variables expectation is zero. then it will be said that. It's a real zero mean randomly.

That's it. >> PRAKAASH A S: Okay, on the four year, if you take e. how inside the bracket Expedition of X minus expiration of expectation of X. Has come. No. >> MA 1004: See if I break down this, it will be expectation of x minus expectation of >> PRAKAASH A S: expectation of expedition of >> MA 1004: expectation. Right. >> PRAKAASH A S: That's yeah. >> MA 1004: Yeah, it will be this. and this is a constant term. Let's let us denoted as C. So >> PRAKAASH A S: Okay. >> MA 1004: expectation of C is >> PRAKAASH A S: See. >> MA 1004: so, this will be equal to expectation of x minus expectation of x, only Yeah.

>> PRAKAASH A S: Okay, C. Minus e, 0. >> MA 1004: Yeah. so like if this is C and this is get cancel, it will be 0, right? >> PRAKAASH A S: Okay. >> Pronod KUMAR BHARATIYA: and right here, working this Expectation of facts, minus expectation of X, as expectation of X, minus expectation of X, I'm not getting this, and I'm unable to digest this one because whenever we are within a set of data is supposed there are We find. It's me. That is expectation of X. So first, we will find the difference between every point of x. With that mean. Some will be positive. some will be negative and when we add them it is always equal to 0. And why we have to break this one to explain this. What to prove, >> MA 1004: See. >> Pronod KUMAR BHARATIYA: This because it is where take any >> MA 1004: See. >> Pronod KUMAR BHARATIYA: data set. >> MA 1004: Listen see. >> Pronod KUMAR BHARATIYA: even if it is. >> MA 1004: Yeah, I got it.

I got it. See zero mean, randomly. random variable states that if you are given a random variable, it's okay with expectation of that random variable at 0, we are getting that expectation. The average value is 0. Then it this variable is known as the zero mean random period. Fine. >> Pronod KUMAR BHARATIYA: Remember expectation of any variable, whether it is random or non-render, then we find the expectation of a meat given series. it will always be 0. >> 24F2003139 SHRUNGEE CHANDRASHEKHAR BHAVSAR: man, can you take? >> Pronod KUMAR BHARATIYA: The.

Difference between, I mean, the difference what we are doing here? >> MA 1004: The expectation can't always be 0. >> Pronod KUMAR BHARATIYA: No, no. I am not expectation. The defense between that >> MA 1004: Then. >> Pronod KUMAR BHARATIYA: particular Ah, Of that particular. Series of X. >> MA 1004: Yeah. >> Pronod KUMAR BHARATIYA: oh, here you are telling that if any single random variable is given with its expectation is equal to 0. It's X is not a series. >> MA 1004: You know. >> Pronod KUMAR BHARATIYA: X Independent is a single variable. >> MA 1004: Yeah. >> PRAKAASH A S: None, which is a 0, mineral yrx here. You are defining the definition for. >> MA 1004: I'm just defining by as a random variable here. And why is a function, like I am denoting by as x minus expectation, x? this is, >> PRAKAASH A S: Okay, why is called? a 0 mean random variable? >> MA 1004: Yeah. >> PRAKAASH A S: virtually? >> MA 1004: So this here like it's it's like obvious. if I'm there Taking the expectation of this, the value will be zero only right if I'm subtracting any variable by its expected value and taking that expectation, it will be zero.

>> PRAKAASH A S: Yeah. >> MA 1004: Yeah, that's the thing. See. >> PRAKAASH A S: ah, >> MA 1004: the expectation of x is some value. that is, let's say a right. So if i Take this value here. So it will be expectation of x. minus a equals to 0 this can be written, as expectation of x minus a equals to 0. >> 24F2004702 RAVI SHEKHAR: It's not necessary that X should take that value. a in first placement. >> MA 1004: No, I'm just just taking A as a, as a variable like any, like, any value.

Take an example, that's it. For taking any value. Particular values. Just an example, like the expectation is a Let's consider it as a Right. So, if >> Pronod KUMAR BHARATIYA: That means we have calculated the mean of that X as a >> MA 1004: You yeah. so, if I subtract it by that random variable and take that expectation, it will be zero. Only correct. Yeah, that's the thing. So I just subtracted that variable by the expected value. and if we take that expectation, we will get zero. only that's it. Yeah, everybody got it. >> 24F2004702 RAVI SHEKHAR: No, ma'am. Getting confused. >> MA 1004: Just remember that the zero random variable is that random people who's very expectation of that variable is 0. That's >> 24F2004702 RAVI SHEKHAR: Okay. >> MA 1004: Yeah, that's the thing. Here, I just took an example of why as this random variable. And I just calculated the expectation and I got zero as the So expect that is the why is a zero mean random? Meaning That's it.

I just took an example of bias. This, it is a function of x. Okay. and I just calculated the expectation and I got the one you zero. So why is a >> P Baskaran Nadar: May I add something. >> MA 1004: Yeah. >> PRAKAASH A S: Zero mean, and variable. >> MA 1004: Yeah, that's it. >> PRAKAASH A S: because, >> P Baskaran Nadar: Can I can I ask something to clarify? >> PRAKAASH A S: That's why we call it as why. he says zero mean random variable. if you take a expectation for X minus exploitation of X, it becomes 0. If any time every anytime, if you expectation expectation. expiration of X is that value only will come expedition of x value only Will come. so that. >> MA 1004: This. >> PRAKAASH A S: average of average, of average, of average, is only one value. >> MA 1004: Yes. >> PRAKAASH A S: Is it correct, man? >> MA 1004: somebody wanted to? Yeah, somebody wanted to >> P Baskaran Nadar: Yeah, I will.

>> MA 1004: add something. >> P Baskaran Nadar: I will give a simple explanation why we do this. if we take a x to be three numbers, 2024 2013-2034, If we got to find a variance and various other, Complex calculations on that it is going to be difficult manually. if we find the mean of it and subtract and define a new variable, y it will be 0 centered. It's easier to calculate. and then we can go back to the original x, whichever way. So defining a new variable y. as x minus expected, value of x, makes it convenient for calculations.

that was a purpose of introducing a zero mean value. Very >> MA 1004: yeah, as a standard you can say, >> P Baskaran Nadar: That's a normal practice. >> MA 1004: started notion. >> P Baskaran Nadar: One of the practice, basically shifting x to y. We are defining a new variable y. >> MA 1004: Yeah, we are just translating it by the expected value. That's >> P Baskaran Nadar: the reason why we are defining, why is it is more convenient to do work on why and get back to the results to the X domain later. as an example, you can simply take three numbers 2024-2040 2017 or something like that. If you go to do the calculation of a variance you will find it. They are very large number to handle is complex. if you shift it to zero mean, It's easier. Smaller numbers. There are some convenient properties based for the zero mean, using that we can find all those properties. Then shift back to x.

It's one technique. that's all. >> MA 1004: Is it clear now? >> 24F2004702 RAVI SHEKHAR: Yes, ma'am. >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes, ma'am. >> MA 1004: let's go into other sanction, which is various. Okay. So in expectation, we calculated the Okay. so why we calculate the variance? We calculate the variance, to know the spread of that data, right? So for example, Like x is taking value. Access taking only one value as 10. Y is taking only value. 9 comma, 11, and Z is taking value. 0 comma. 20. Okay, these are the values. So if i calculate the expected value of everyone of x, y and z, we will get as 10. Okay. So, all the values of expectation are same for all the variables. Okay. but this like like if you look into the data that data of x is different, the data of why is different. and data is that is different.

So we are not able to conclude about the data from the expected value, right? So for that, we calculate the variance of x variance of y and variance of them. Right. to to see about like to visualize about the data. How much spread is there in that data? So if you look, that 10 is just one point, right? From 9 to 11. There is not that much of spread thick Two values. Different. is that from 0 to 20, it's a spread, right? zeros here and 20 is here 9 and 11. That's it. then. So if you look the spread if you have to visualize the spread we calculate the wheels Fine.

Is it clear? Why we calculate the variance? Yeah. >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes. >> MA 1004: So, the formula variance of X's. expectation of X minus expectation of x whole square. This is the formula of valence, and if we have calculate the standard, Deviation of x, it will be the square root of variance of base. Is this formula here? Yeah. So if we have, if the values of variance of x, like like the variance of X is more that means the spread is more. Right. so like, for example, Like this example. x is taking 10 value y is taking line and z. So if you have to calculate the variance of x, Right. So variance of X will be. That is expectation of. What is x? That is 10 minus expectation of x is Then whole square, right? This will be 0.

This is clear. how I calculated the variance. For the x variable. Right. >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes. >> MA 1004: For the BI variable, it will be. expectation of That is a expectation of these values. so it will be equal to That is 9 minus 10, whole square with probability. Like, if this is the probability, they are taking one by two and one. 1 by 2 plus 11, minus expected, value whole square. multiplied by 1 by 2. Is the step here. Is. >> 23F3004210 CAESAR PARTHO KARMOKAR: No ma'am. >> MA 1004: No. Okay. See the variance of x is calculated by expectation of x minus expectation of x composed square, right? so, here the values of the values of by the y are taking two values, 9 and 11. Right. so, if you have to calculate the values of x, like if you have to calculate the expectation, simply, it was 9 into 1 by 2 plus 11 into 1 by 2.

This is how we calculated the expectation, right? >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes. >> MA 1004: Now, we have to calculate the expectation of x minus This. So, it will be. What is X here? >> P Baskaran Nadar: Way. they will get confused. Why minus the expected value of y? >> MA 1004: Sorry. Okay. so the value of y is What is the value of y? >> 23F3004210 CAESAR PARTHO KARMOKAR: Nine. and 11. >> MA 1004: 9 and 11, right? So for the first, Nine Minus. What is the expectation of I here? The expectation of y is 10 So 9 minus 10, whole square multiplied, by the probability.

That is One bite. Plus the other value of y that is 11 minus expectation. that is 10 whole square. multiplied by 1 by This is how I calculated this. Is a square now. >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes. >> MA 1004: Sure. >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes. >> MA 1004: So if I calculated if I calculate this value, I will get one as the answer. Similarly, if I calculate the variance of z, it will be equal to 0, minus 10 whole square multiplied, by 1 by 2 plus. 20 minus 10 whole square multiplied, by 1 by 2. Okay. So if I solve this, I will get 100 as the answer. So if you see the expectation of all the random variable will see for the variance is different for x is 0 for y is 1, and for z is 100, right? So for that, the variance is hundred That means the spread is more.

Compared to why. So why? for why is it is 1? So the spread is okay, fine. And for the x, it is zero. that is there is no spread. Got it. >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes, ma'am. >> MA 1004: Yeah. So this is why that is why we are calculating the values. To know the spread of the data. Because from the expectation, we are not able to know or it will be not able to visualize about the things about the data. That's Fine. >> Pyari Singh K: This number has no mathematical significance. In quantifying the spread, is it? >> MA 1004: So, what? >> Pyari Singh K: I mean, the number. and we cannot actually make out how much is the spread we can only say relatively.

Is that what you're coming to? >> P Baskaran Nadar: Aviation will give you that standard deviation will give you more Clarity because you take a square root, it becomes meaningful. From the variance, you take the square root of it. >> Pyari Singh K: Oh yeah. >> P Baskaran Nadar: It comes. the standard deviation that will be related to the mean. so that you know how many sigmas you are away. That will come later. >> Pyari Singh K: Right. Yeah yeah. >> MA 1004: Bye. So like, if you have to calculate like, Exits. taking values. One common. Six. So how we will calculate the variance of X? >> Pyari Singh K: We find. >> MA 1004: What is the formula of? >> Pyari Singh K: addition of X minus.

Expectation of X. The whole square. >> MA 1004: Yeah. So, let's like the value. the excess taking value, 1 minus the expectation of Like, in earlier, we calculated that it's friction of axis 3. 1, Right. So it will be 1, minus 3.5 whole square. Multiplied by the probability that is one, right? Similarly, we can calculate the other values. >> Pyari Singh K: Is it one by two or >> P Baskaran Nadar: Is one by. >> MA 1004: Sorry, 1 by 6. Plus three minus 3.5 whole square. multiplied by 1 process. So here we can put all the values to 6, minus 3.5 whole square, multiply 1 plus 6. So, is it clear how to calculate the variance? >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes. Yes. >> MA 1004: Okay. >> 23F3004210 CAESAR PARTHO KARMOKAR: ma'am. >> MA 1004: Now, we have some of the properties of various The first property, says states that variance of A into x. that is if you are multiplying any variable with some constant, it will be a square into variance of X.

The constant will be a square. And we can it is multiplied by the variance of X. Fine. Second, the standard deviation of a into X will be. mode of a into standard deviation of Is it clear? Because like the very standard deviation of excess, the square root of variance of x, right? So the standard deviation of if you are putting a calculating the standard deviation of a of x. It will be the square root of A square into V of X. Correct. and square root of a square is mode of a. We are just taking the positive value and square root of V axis standard deviation of x. That's it. Got it.

Is it clear? >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes. Yes. ma'am. >> MA 1004: Yeah. Then the third properties variance of x plus A Will be. Variance of exponent. okay, we will not take the value of a if you are multiplying. any constant, then we are squaring it. If we are adding any quantity, The variance remains the same. That's it. Okay. And the fourth property, the standard deviation of x plus A will also be equal to the standard deviation of Is are all the properties. The core property. Clear Okay. Now we calculate there is another formula variance of X. Did this expectation of x square minus X expectation of x whole square. Yeah. So, expectation of X square can be calculated as submission of It will be. Square probability of sorry, functional X equal. >> 24F2004702 RAVI SHEKHAR: Feelings. >> MA 1004: to t.

Correct and expectation of X can be written as summation of p probability of x equals to t. Whole square. Correct. >> 24F2005774 ABHISHEK KUMAR: Museum. >> 24F2004702 RAVI SHEKHAR: well, Mom in the latest recruit also be whole >> MA 1004: ah, So here the x square it is denoted, as the This is denoted as a second moment and expectation. affairs is called as the first woman. and variance of X is called as the second central. These are some of the terms that they just have to just just remember that. That's it. X Square is denoted as the second movement. expectation of X, is called, as a first permit. and millions of Texas known as Second Central. Fine. >> 24F2004702 RAVI SHEKHAR: For month. Second central. >> MA 1004: Central. Yeah. Fine. and proceed for the >> 24F1002378 KHEMRAJ DHANRAJ NAKHATE: Yes. >> MA 1004: Yeah. >> 24F2004702 RAVI SHEKHAR: yes, Ma >> MA 1004: now comes the there is one property.

In expectation, if we are given two variables x and y and we are calculating the sum Arabia calculating the expectation of some of two variables. It was >> Pyari Singh K: Expectation of X plus expectation of Y. >> MA 1004: Yeah. In this, it was not necessary that these two variables should be independent or dependent, right? So but if it is given that x and y are independent, okay, that expectation of x into y will be equal to expectation of x into expectation of what This is the first case, if they are independent and the second case the variance of x plus y, Will be equal to variance of X plus variance of Y. These are the two conditions. if x and y, are independent, >> Pyari Singh K: so, if it does not independent we cannot sum the >> MA 1004: We cannot know. only if they are independent, then we only we can calculate the variance of sum of these two. Clear. Okay. Now, there are some distributions, Okay for bernality distribution. The variances. P into 1 minus p. Right. How we calculated it? See the >> P Baskaran Nadar: Npnp EN p 1 minus p.

>> MA 1004: I'm talking about Burnley. >> 24F2005774 ABHISHEK KUMAR: I know. >> MA 1004: But knowledge. See. in Bernoulli, the X is taking value 0. Comma 1. Okay. And the probability is P and this is one minus p. So, if you have to calculate the variance of X, It is expectation of X square minus x. of X whole square. so, Expectation of X square will be. >> Pyari Singh K: Minus B. 1 Minus B. >> MA 1004: Expectation of X square. >> 24F2004702 RAVI SHEKHAR: So yeah, 1 minus p. >> Pyari Singh K: 1 square into 1 minus B, right? >> MA 1004: 1. Yeah. 1 square into 1 minus. It will be part of what minus. and expectation of, X will be.

>> Pyari Singh K: B. >> 24F2004702 RAVI SHEKHAR: B. >> 24DS3000097 SOURABH RAJEEV MUNGARWADI: History. >> P Baskaran Nadar: EP. I don't know why it is. 1 minus B. >> Pyari Singh K: the problem is, >> MA 1004: Just a second. No, no. >> Pyari Singh K: you. Yeah. >> MA 1004: See the probab. Yeah. Different. >> 24F2004702 RAVI SHEKHAR: you have taken probability different. >> Pyari Singh K: To be. >> MA 1004: Yeah. So 0 is taking probability, 1 minus b. and 1 is taking the probability. Right. Then this will be equal to >> Pyari Singh K: P squared. >> MA 1004: This will be. And this will be. To get that. So if I put this value in this, so it will be equal to expectation of x squares.

P minus expectation of x whole square will be P square. so taking P as a common, it will be into 1 minute speed or it can be private as p into Q. Right. >> 23F3004210 CAESAR PARTHO KARMOKAR: Then we'll do please. Explain how did you get expectation of excess square? >> MA 1004: Simply as like expectation of X square will be 0, square multiplied by 1 minus p. Plus. 1 square multiplied by its probability. that speak. >> 23F3004210 CAESAR PARTHO KARMOKAR: Okay, clear. >> MA 1004: Yeah. Yeah. So veins of banol is clear. >> Pyari Singh K: Yes, ma'am. >> MA 1004: So, you have to calculate the variance of binomial. so, it will be equal to variance of X1. plus x 2 till x in and all are independent.

Then it will be equal to variance of x 1 plus variance of x 2. Then x n, right? For one variable, the variance was p q, right? So, it will be p q. plus p q. so there are n terms, it will be any or you can write it as n. P, 1 minus. Is it clear? Now comes the standardize random variable. What is the standardized? Random variable, a random variable x is a standard. If the expectation of x is 0 and variance of excess 1, In a random variable x is said to be standardized. If the expectation is 0 and variances one, is it clear? >> 24F2004702 RAVI SHEKHAR: You please repeat once now.

>> MA 1004: See if a random variable that x is random variable, okay? We have to say that it is standardized when expectation is 0 and variance is one. then x is said to be standardized. Okay. and if I have to standardize a random variable, how will I like if Y is a variable and I have to it, then I will. like, if I have to convert this variable into standardized rate, random variable, then I will calculate it by x minus expectation of x divided by variance of x. like if x is a random variable, if i subtracted by the expectation and divided by the variance, then I will get a standardized random video. >> P Baskaran Nadar: It divided by Sigma not variance. >> MA 1004: So yeah. >> 24F2004702 RAVI SHEKHAR: Square root. Home. And basically what we are doing is we are shifting. From X to y. So that it becomes a standard. >> MA 1004: Yeah, just we are shifting. the x variable. We are converting into a standardized random variable. It will be easy for the calculation. that's it.

That's why we standardize randomly. >> 24F2004702 RAVI SHEKHAR: Yes, ma'am. >> MA 1004: So, till now till the variance and standardize, everything is clear. Is it clear? Everyone. >> 23F3004210 CAESAR PARTHO KARMOKAR: Us1. >> MA 1004: Yeah. Now comes the covariance. Okay, so what is covariance? if you want to? summarize? Any relationship between two random variables effects and wire two random variables. Okay. And if I have to summarize the relationship.

What is the relationship between the two random variables? That I calculate. the cool wheels. Like, if I have to know if the values of x is increasing, then how it is. affecting the values of money, right? So if I have to like, if I have to know, or summarize the relationship between these two, then I calculate Okay. and the formula of covariances expectation of x minus expectation of y multiplied by expectation of y, y minus expectation of what? Clear. Is it clear? Why we calculate the covids and the formula? >> Pyari Singh K: Yes, ma'am. >> MA 1004: Yeah. So, the first thing if the covariance is positive, Okay. That means if X is increasing, then why is also increasing? secondly, if coincidence is negative, that means if x is increasing, then why is decreasing That it's going opposite. Okay. And thirdly If it's zero. Right. So that mean it is uncorrelated Is this clear? Everyone. >> JB ANMOL: Can you give us an example? Like what do you mean by when X is increasing? Or why is it >> MA 1004: yeah, just like, for example, ah, Right.

It is a height of a person, okay? And why is the weight So, if a person height is increasing then the weight is also increasing. Right. If a person is having a like, These increasing his height, the the weight is all gets increase, right? For the normal general cases. >> JB ANMOL: Yeah. >> MA 1004: So, the covariance is positive right That means if x is increasing, that means the y is also increase. if I take another example here, like, X's you can say, Orce. the runs. by a cricket, like a person by and why is the wicked? Is increasing.

That is the runs. There's scoring the runs in the wicket will be decreasing, right? The number of tickets will get decreased their scoring. number of runs. if a one person is scoring a huge number of grants, then the wicket will be decreased, right? So, it's negative. >> JB ANMOL: Okay, ma'am. in terms of random variable like how will I define at? Let's say I have a random variable 1 to 10. How will I define that if this is increasing? or this is decreasing >> MA 1004: X x is taking 1 to 10. >> JB ANMOL: Yeah. >> MA 1004: And bye. >> JB ANMOL: And why is taking let's say 10 to 15. >> MA 1004: Yeah.

Okay. So Like, you will be given a joint. distribution of that x and y. Okay? >> JB ANMOL: Okay. >> MA 1004: Then we will calculate the covariance. And we will see if we are getting the negative value, positive value or what value? you're getting, if you are getting the positive value, that means the the values of x is increasing in the value for us if you are getting negative value that means it exile is increasing. Y is getting Just the calculating, the Kumar is then only we can get the relationship of the etc. >> JB ANMOL: Can this be possible? Like if the covariance is positive X can be decreasing in us, Why? I can be decreasing as well or it's just increasing only >> MA 1004: Sorry.

>> JB ANMOL: The first case, man. when covariance is positive. can X and y both decreasing as well. >> 24F2004702 RAVI SHEKHAR: That will be positive correlation. >> P Baskaran Nadar: This this both album moving in the same direction. is Ovariance is positive. >> JB ANMOL: Okay. >> P Baskaran Nadar: If they both move opposite direction it is negative. If this movement is insignificant that is closer to zero than they are not correlated. >> JB ANMOL: Okay, thank you.

>> Pronod KUMAR BHARATIYA: Man, doesn't you explained this? positive negative and zero in the context of correlation Because what? I have understood that ovariance, simply means that what's the direction of x and y. Whether x is moving x and y's moving in. Same direction, x and y is moving in opposite direction and excel. By is not moving in any similar direction. and correlation means that the variable acts and variable.

I variable by our positively correlated means that they are moving in. Same direction. Negatively correlated means they are moving in opposite direction. as when correlation is 0, then there is no relation between the movement of the variables. >> MA 1004: See in correlation, we calculate the strength. The correlation values are between minus 1 and 1, right? >> Pronod KUMAR BHARATIYA: Yes, yes.

>> MA 1004: So if you are having the values is like >> Pronod KUMAR BHARATIYA: 0.5. >> MA 1004: Minus. Yeah, minus 0.5 or something like that. So from that we get the strength of the relationship between them >> Pronod KUMAR BHARATIYA: Now, how strong? >> MA 1004: If it's >> Pronod KUMAR BHARATIYA: x and y are correlated. >> MA 1004: Yeah. Yeah, me in this. We are only getting the direction that is, but >> Pronod KUMAR BHARATIYA: the direction, whether axis >> MA 1004: that we are getting the strength and the direction. Fine. Yeah. Let's take an example of how to calculate. okay, so if I'm having Ahmed joint distribution table, as Physics. Taking value, minus 1. And 1. And y is taking values as minus 1 0 Probabilities are 1 by.

50. So how will you calculate the covidiance of X and y? It will be equal to expectation of X. Minus expectation of x. and y, minus expectation of so first of all, here, we have to calculate the expectation of X and expectation of Y Correct. So how we will calculate the expectation of X. Any idea. >> Pyari Singh K: finally possible value, sex can take >> MA 1004: C. Yeah. Okay. >> 24F2004702 RAVI SHEKHAR: Oh, my music. >> MA 1004: so we will find >> 24F2004702 RAVI SHEKHAR: marginal. >> MA 1004: Marginal. Yeah. if x will be >> P Baskaran Nadar: Just a minute. Just a minute. Your probability doesn't add up to one. You may have to tweak those figures. Your joint probability. Distribution doesn't add up to one. I, Also worry, sorry. >> 24F2004702 RAVI SHEKHAR: It's coming out to be one.

>> P Baskaran Nadar: Sorry. Sorry. Well, my mistake. >> MA 1004: Yeah. so we were calculating. So we will calculating the expectation of X. So first of all, we'll find out the marginal distribution. okay, so for minus 1, the marginal distribution is 5. >> 24F2004702 RAVI SHEKHAR: 550. >> MA 1004: That is 1 by 3. Similarly for 0, it will be 1 by 3 for 1. It is 1 by 3. okay so the expectation is Minus 1 into the probability for the function. Plus 0 into 1 by 3 plus what? one into one. Correct. Is this clear? >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes, ma'am. >> MA 1004: So it will be 0. Similarly, we can, we will calculate the values of y. So what is the marginal distribution of FY? for minus 1, it is >> Pyari Singh K: One by three. >> MA 1004: And fy will be 0. so, these value Is 0. So at last we will get the ovariance of x, comma y, s expectation, of x and y. Right. so, expectation of xy will be That is for minus 1.

Into minus 1. The probability is 1 by 15. Similarly, minus 1 into 0, the probability is >> 23F3004210 CAESAR PARTHO KARMOKAR: 2 by 15. >> MA 1004: really minus 1 into 1 because so we will calculate all other values. Right. So if we calculate these value, we will get as minus 2 by 50. Okay, here we are. getting the negative one. The covariance is negative. Right. So it is like we can conclude that if x is increasing, then y is decreasing.

Or if y is increasing, then x is degrees. That's it. They are going in opposite direction. Fine. This is how we calculate. Is the calculation part here? >> Rahul: Since we have the same values that x and y can take. and they also have the same probability. I'm finding it hard to imagine how. They move in the opposite direction. What does movement even mean in this situation? >> MA 1004: Take its like Newman in the sense. if the values of x is increasing the values of wise, decreasing like, sort of like that. It's just an example. that's it. So if >> 23F3004210 CAESAR PARTHO KARMOKAR: Then we should be minus 1 by 15, right? Not minus 2. >> MA 1004: It's coming out to be immediately >> P Baskaran Nadar: I think E x y calculation. >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes. >> P Baskaran Nadar: We have only four combinations to calculate. I think we can calculate Right. >> MA 1004: Yeah. >> P Baskaran Nadar: We can, we can >> MA 1004: Yeah, begin. >> P Baskaran Nadar: strike down the middle row and middle.

column just four. Colum four, four elements left. >> MA 1004: yeah, you >> P Baskaran Nadar: That seems to be something. >> MA 1004: So people. Plus. >> P Baskaran Nadar: Right. Almost a 0. >> MA 1004: Minus 1. >> P Baskaran Nadar: They are not related. >> MA 1004: Sorry 1. :15. Getting positive. This is clear. So, again emissions. >> Pyari Singh K: Are the value seems to be 0. >> MA 1004: We will get -1. so, we are getting -1 >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes, ma'am.

-2 by 15. >> MA 1004: so, it's just that it's just showing that if the values of x is increasing, then the values of white Decreasing, all white servers? That's it. Like, they are going in opposite direction. Is it clear? how to calculate the covariance? >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes, ma'am. >> MA 1004: And anybody having any doubt? Canon Porcito. The calculation part is clear. >> Rahul: Yes. >> MA 1004: so, there are some properties of covariance as well. So, if I have to calculate the covariance of x comma x, Right. So it will be variance of X. Right. The capital calculator, the same variable, then it is equal to the variance of that variable. Okay. And secondly Ovariance of X comma, Y can also be calculated as expectation of x. y, minus expectation of X into expectation of Okay. Thirdly. Covariance of. x comma a by Plus B z. is calculated as ah, Plus b into covariance of x comma z. Okay. and if we have Covalence of Ax Plus b y comma Z.

Then it will be calculated as A covidians of x comma z plus b covalence of y comma z. Are these properties here? Yeah. If x and y, are independent. okay, that independent then covariance of X and y, are 0. Fine. That means they are uncorrelated. So if two random variables are independent, that means that they are uncorrelated Okay. but the reverse is not, that is if it is given that they are uncorrelated. that doesn't mean that in their independent, they can be dependent jobs. Fine. Is it clear? >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes. Yes. ma'am. >> Rahul: that really be the case if x and y are Not affecting each other is in that enough to say that their independent.

>> MA 1004: Oh, there are people. It is not necessary. Like, for example, we can take an example. X and y. They are taking values. – 1. 0 1, and why is taking value? See you. About these are 1 by 8. 1 by 4 1 by so, first of all, like, We can, if we like it is given that the covariance is 0 here. We calculate the COVIDENCE of excess, we will get that it has 0, okay? that means it is uncorrelated Now, if you have to find out that like the dependence, So it will be. we know that if x and y independent, the expectation of x comma x into y, is expectation of x and expectation of y.

>> 23F3004210 CAESAR PARTHO KARMOKAR: Yes. >> MA 1004: so, No, sorry. we are calculating the independence of these variables, right? So if you have to calculate the independence, we will take the values of x y as function of x. and function of, right? So if I have to calculate, Like, for x is taking value. Minus 1 and y is taking value. Oh, what? Right? For this. The value is 1 by 4, correct? If x is like taking value, minus 1 and 1 is take x is value.

Minus 1 and y is taking 1 the probabilities one by four, right? but if i calculate for x is equal to minus 1, it is Final distribution. It will be three by Eight. Okay. and for y, is equals to 1, the marginal distribution is >> Rahul: 2 by 4. >> MA 1004: Yeah. >> Rahul: Or for it. >> MA 1004: So it will be 1 by 2. So if we multiply these values, We will get. 3 by 8 into 1 by 2, which will be equal to 3 by. 16, right? So here the values, the joint distribution of fx. And if we multiply each, they are not equal, right? >> Rahul: Yes, but >> MA 1004: So that means they are not. independent. they're dependent Yeah. so that's the case. It is like we have been given the covid is 0.

That means they're uncorrelated. but if we find out the dependency, they're not dependent. >> Rahul: Okay. >> MA 1004: Sorry. they're dependent, they're not independent. So always. >> Rahul: Yes, ma'am. >> MA 1004: remember, the case, the reverse case is not always true. If it is given that their independent then, that means they're uncode, that's for sure. But the reverse is not compulsory if it is given uncorrelated that doesn't mean that there dependent that that independent they can become. >> Rahul: This one. >> MA 1004: Got it. >> Pronod KUMAR BHARATIYA: Woman. We can feel like this also but if X and y, random variables are dependent.

And covariance between these x and y, may or may not be equal to 0. >> MA 1004: Like if they're dependent and covariance. >> Pronod KUMAR BHARATIYA: Covariance of that. the random variable x and y, may or may not be equal to 0. >> MA 1004: It will not be equal to 0. >> Pronod KUMAR BHARATIYA: If they are dependent. Okay. >> MA 1004: Ah, T. Calculated. As expectation of x y minus expectation of x into expectation of, right? And we know that if two random variables are independent, then expectation of X y's Equal to expectation of Y. Correct. That independent. >> Pronod KUMAR BHARATIYA: Yes. >> MA 1004: Needs. These two values are equal. they will get canceled. It will be equal to 0. >> Pronod KUMAR BHARATIYA: Yes. >> MA 1004: And you're saying that they're dependent if they're dependent, then this condition will not be satisfied.

>> Pronod KUMAR BHARATIYA: Yes. >> MA 1004: If this condition is not satisfied, we will not get the covenants as you. >> Pronod KUMAR BHARATIYA: it will in case of dependence between two random variables, their co-varians should never be equal to 0. >> MA 1004: know, because It's two values are not getting equal. >> Pronod KUMAR BHARATIYA: Yes. Okay. >> MA 1004: Yeah. >> P Baskaran Nadar: I think. >> 24F2005617 SWARNAVA CHATTARAJ: Did you get the above example? >> Pronod KUMAR BHARATIYA: It was. >> MA 1004: Okay. >> Pronod KUMAR BHARATIYA: for independence case and considering dependent.

>> MA 1004: Calculate the covariance of this, we can calculate if you want. Should I calculate? >> 24F2005617 SWARNAVA CHATTARAJ: No, no. >> MA 1004: Yeah. >> 24F2005617 SWARNAVA CHATTARAJ: I'm not asking you to calculate, I just asking what was the assumption here that we first assumed? >> MA 1004: No, you'll be just I'm taking the example if we calculate the covariance of this. We are getting covenants as 0, I have a calculated it after calculating and getting covenants as 0. Okay. And if I calculate the dependency or the independency of these two variables, it is getting dependent. Right. Because fx is not equal to fx into F, right? >> 24F2005617 SWARNAVA CHATTARAJ: Okay. Okay. >> MA 1004: Independent. Yeah. >> 24F2005617 SWARNAVA CHATTARAJ: so if the covariance is 0, it does not directly imply that they will be always independent. >> MA 1004: Correct. Okay, it is. it's just says that if x and y are independent, Then it is always true that the core variances 0, that is their uncore. But the reverse is not true, like it is not necessary that if you have been given, right? If you have been given that coincidence, 0, that doesn't mean that these two variable will always be dependent.

They can be dependent So it is not always true for the inverse things. but this case, it is always true that it is given expanding dependent. that is coincidence 0. like, without any thought if it is given independent, that means The coincidence is you, but the reverse is not true. Right. >> Pronod KUMAR BHARATIYA: And if covariance is not equal is equal to 0, if this doesn't mean that x and y are independent, It may be dependent. So then it is dependent. So when we say that if x and y is dependent, And covariance should never be equal to 0. We concluded that earlier. >> MA 1004: Yeah. >> Pronod KUMAR BHARATIYA: but in this case, >> P Baskaran Nadar: Well, one minute.

Independence decides whether the covariant can be 0 or not. Not the other way around Ovariance Does not give Anything to about independence. Yes. >> Pronod KUMAR BHARATIYA: Dependence or Independ. >> P Baskaran Nadar: There can be a situation where they are dependent. >> MA 1004: Independent. Or dep. >> P Baskaran Nadar: And still covariance is zero. >> MA 1004: Yeah. >> P Baskaran Nadar: That can be a sample that maybe the sample is such, we get a covariance zero. >> Pronod KUMAR BHARATIYA: Ence. but they are dependent.

>> P Baskaran Nadar: Independent. The independence. Is that structure itself? the experiment or whatever? The that is the experiment. itself. Whatever. >> Pronod KUMAR BHARATIYA: so, >> P Baskaran Nadar: we are doing. A covariance is an inference from the data that we observe. >> Pronod KUMAR BHARATIYA: So suppose if covariance is equal to >> P Baskaran Nadar: Yes. >> Pronod KUMAR BHARATIYA: 0, this doesn't imp that the variables.

>> P Baskaran Nadar: Yes. >> Pronod KUMAR BHARATIYA: the random variables which we are studying, they are or independent. >> P Baskaran Nadar: Correct. >> Pronod KUMAR BHARATIYA: So, this means that if covariance is >> P Baskaran Nadar: Hmm. >> Pronod KUMAR BHARATIYA: equal to 0, then our random variables, the samples, which we have taken for that particular random variable can be dependent can be independent. >> P Baskaran Nadar: Yes, it doesn't. >> Pronod KUMAR BHARATIYA: yes, this is >> P Baskaran Nadar: talk anything about independence or dependence. >> Pronod KUMAR BHARATIYA: We cannot conclude the by simply the seeing the, or calculating the covariance physical to zero. >> P Baskaran Nadar: You are absolutely right, Covariance >> Pronod KUMAR BHARATIYA: Yeah.

>> P Baskaran Nadar: does not give any >> Pronod KUMAR BHARATIYA: Imp. >> P Baskaran Nadar: information about the independence or dependence. >> Pronod KUMAR BHARATIYA: Independence, or >> P Baskaran Nadar: whereas the once, you know that that dependent or independent? If it is independent, we know very clearly covariance has >> Pronod KUMAR BHARATIYA: That covariance will be zero. >> P Baskaran Nadar: You need not calculate. very waste your time. It's >> Pronod KUMAR BHARATIYA: so, >> P Baskaran Nadar: No reverse. >> Pronod KUMAR BHARATIYA: so sir, I am thinking in the reverse sense. >> P Baskaran Nadar: That's what she keeps saying. >> Pronod KUMAR BHARATIYA: And no, no, no, no, no.

And reversing that sense. Not if we know that x and y are dependent, >> P Baskaran Nadar: Yes. >> Pronod KUMAR BHARATIYA: Then in that. case, covariance will not be equal to 0. >> P Baskaran Nadar: Not necessary. >> Pronod KUMAR BHARATIYA: That I am and trying to understand. >> P Baskaran Nadar: It can be, it can be anything. >> Pronod KUMAR BHARATIYA: It can be zero or it cannot be. >> P Baskaran Nadar: Correct. >> Pronod KUMAR BHARATIYA: That that I am trying to understand. >> MA 1004: Is a piano. >> Pronod KUMAR BHARATIYA: Yes. >> MA 1004: The covenant part is clear. Now comes the correlation part. So, what is correlation in covid? Is we got the direction, but in correlation, we get the strength of the of the two random variables, right? So what is the string between the two render variables? How? like, how positively correlated are both of the random variables or how negatively correlated Botha, okay? So the formula of it is denoted by row of x and y And the formula is ovariance of. X comma, y divided by the standard deviation of x multiplied.

By the standard deviation of Okay. And the values. the ranges, the correlation ranges from minus 1, 1. Okay? So if the correlation is minus 1, that is it is strongly negative if it is one, that means it is strongly positive. And if it is zero, that means it is uncorrelated Is it clear? >> Pyari Singh K: Less than or equal to right? >> MA 1004: Yeah, that's that. >> Pyari Singh K: Okay. >> MA 1004: Listen. Yeah. Listen. No, it's Yeah. it's less than if it's minus 1, then it's strongly negative. If it's one, it's strongly positive. Yeah. >> 23F3004210 CAESAR PARTHO KARMOKAR: Clearman. >> MA 1004: So, if we like, This may be calculate the correlation if we want to find out the correlation through a graph. Okay. So like if I draw This is the y value and this are the xy. okay, so if I draw manually like this, so, these data points are like The.

value one. And this is taking three. So this point is One. common. Similarly, all the data points are like. so here we if we see that it is not showing any trend, right? It is not like it is not showing if it is increasing or decreasing or anything, right? is it like it is uncool it. like, if the values of x is increasing, there is like, we cannot say the values of y is also.

Some of it is like X is increasing The Y is decreasing and some is like X is increase. It is increasing wiseauctions. So, we cannot say about the relationship between the x and y, right? So, the correlation is 0. Is it clear? Yeah. And if I'm having another graph, Like this. >> Pronod KUMAR BHARATIYA: Positively. >> MA 1004: So, this is positively correlated. Right. That means if s x is increasing the Pi is also increasing. Correct. >> Pronod KUMAR BHARATIYA: because, >> MA 1004: And if I have this type of graph, >> Pyari Singh K: Negative one. >> MA 1004: Then. It's negative. that is if x is increasing, the y is getting. Decrease. so, from the graph, you can conclude if what is the relation between the correlation between the Excel part. If it is strongly positive.

Strongly negative or like that. See this is close to one because it is linear. Right. So this will be plus one that is this and this is also linear. Right. So this is also close to minus one. so it is strongly positive and this is strongly negative. If I'm having a graph, Like this. let's say. It's not a bit linear, but it's like it's showing that it's the values of X is increasing and the values of y's increase.

Like it's not linear, right? That means it is not. It is close to like we can say, >> Pyari Singh K: Point. >> MA 1004: Yeah. means it is not that it is not one but near to that's it. Between zero and one that it is positive. That's Similarly, we can have the graph like this negative direction, then it will be between minus 1 to 0. Right. >> Pronod KUMAR BHARATIYA: and if it's a straight line upward or downward, then it will be perfectly for positive or perfectly negative. >> MA 1004: Yeah, like it's linear if it's linear like this. >> Pronod KUMAR BHARATIYA: Yeah. >> MA 1004: Fine. >> Pronod KUMAR BHARATIYA: so, the width of the scatter of the, points, xy coordinates that defines that how much strong it is on the negative side or on the positive. >> MA 1004: Hmm. >> Pronod KUMAR BHARATIYA: Side.

>> MA 1004: Yeah. >> Pronod KUMAR BHARATIYA: You can conclude like that. >> MA 1004: Yeah. >> Pronod KUMAR BHARATIYA: If the depth of the scatter is very minimal, then it is very close towards one. if it is white white and we can Moving between anywhere between 0 and >> MA 1004: Is this black here correlation is clear? >> Pronod KUMAR BHARATIYA: Yes. >> MA 1004: Now the last topic is Markov's, inequality and turbitions So what is Markov's inequality? Markov's inequality states that if x is any random variable, okay? taking non-negative values. if x is only taking non-negative work, just remember that with finite you here. Mu is the mean of that randomly. Okay, we need not me mean as mu, right? So, if x is a non-negative value, not variable.

And with a finite mean that is real, then probability of X greater than equal to C. Is always less than you by C. This is the mark of inequality. In like this will provide the upper bump. Right. So this is the greater than sign. So it will provide the upper bound. that a non like that. this value is greater than equal to some Positive. Fine. >> Pronod KUMAR BHARATIYA: Is repeat. Once again this E x is greater than equal to C. >> MA 1004: see if see the mark of inequality is just that it tells us We can say that like how this X random variable is far or larger than the mean guy. that's it. okay, so, if x is a random variable taking only negative values with the finite mean, then the probability of greater than C is always less than equal to new upon C, but use the Me, you have to just remember this function.

>> Pronod KUMAR BHARATIYA: Presented Batman. It will always If we take, suppose more x and suppose any value x is non-negative and c is also some non-legating. And x is greater than or equal to cease c and the it will be always been less than this issue. Uic. >> MA 1004: Yeah, it will be less than >> Pronod KUMAR BHARATIYA: it will always be this, particularly >> MA 1004: Yeah, this is what expression of Marcos inequality, if we have to find out the upper bump that, what is the upper bound? That the variable is greater than some positive value? What is the upper box like, for example, like Let's take a situation if we if the every number of the average number of minutes, a person, is waiting for a bus.

Okay? It is 10 like for that, that the expectation that he is waiting for a buses then fine. if we have to find out the person. wait more than 30 minutes. Okay. So, what is the upper about, right? So we will calculate probability of x greater than equal to 30, it will be less than you. That is 10 divided. by C was 30 will be this, right? So we are calculating here. The probability that a person has to wait more than 30 minutes, right? So it will be less than equal to 1 by 3.

>> Pronod KUMAR BHARATIYA: You know, what does this? >> MA 1004: The probability. >> Pronod KUMAR BHARATIYA: upper bound and lower bound? >> MA 1004: Sorry, what? >> Pronod KUMAR BHARATIYA: Upper bound and lower bound. What does this mean? >> MA 1004: Like, this is an upper bound. That means the probability that he has to wait. more than 30 it is. Less the probability will be less than one one by three.

That's it. That's the upper after like the maximum point, you can see >> Pronod KUMAR BHARATIYA: It means that if we have to wait for more than 30 minutes, >> MA 1004: Yeah. >> Pronod KUMAR BHARATIYA: Then its probability will always be equal to 1 by 3. >> MA 1004: yeah, less than equal to >> Pronod KUMAR BHARATIYA: Less than all equal to 1 by 3. >> MA 1004: Right. That's the meaning. so, Markov's inequalities calculated when we have the mean. okay, if you have beef, given a random variable, which is one negative and it the mean is given for that random variable. And if you have to find out the upper bound, then we calculate the marcosity one. Fine. Is a Marco's inequality clear for everyone. >> Pyari Singh K: This is the probability he has to wait for more than >> MA 1004: Yeah.

>> Pyari Singh K: 30 minutes. Okay. >> MA 1004: like, if it's, that they are asking the probability, how much for the probability of how much time For example, what was that person has to wait for more than anything. So what is the probability of his meeting? We have to get so, we just calculated the upper one like Maybe it will be one by three or less than 100.

That's it. >> Pyari Singh K: It is not the time, right? It's a probability. He has to wait. >> MA 1004: Yeah, it's the probability. >> Pronod KUMAR BHARATIYA: so, so simply if we consider this example and we have calculated for 30, if we are calculating for 40 it, will be less than 1 by 3. >> MA 1004: Yes. >> Pyari Singh K: Then one by four. >> Pronod KUMAR BHARATIYA: Less than 1 by 4 and 1 by 4 will be less than 1 by 3. Suppose we are calculating that if we have to wait for 20 minutes. 30 minutes and 40 minutes. then the lowest probability will be, which one. >> MA 1004: your calculating for all the 30 or >> Pronod KUMAR BHARATIYA: A different situations. I am considering different. situation. That. the expected, the value of his weight is 10 supposed 10 minutes and what is the probability that is going to wait for 20 minutes? 30 minutes 40 more than 20 minutes more than 30 minutes more than 30 minutes more than 50? Suppose, these are the four or five situations then. I am trying to conclude a thing that which probably will be the least one.

That it will not go. come below that particular. Program. Because here we are saying that it should not be less than or equal to 1, right? >> MA 1004: so, it is the probability is less than equal to. We are just calculating, the upper bound. That's it. >> Pronod KUMAR BHARATIYA: so, it simply means that if the wait time is more than 30, then that probability will in >> MA 1004: Is one by. >> Pronod KUMAR BHARATIYA: any case be less than 1 by 3. Because this is the upper bound up to 30 minutes.

>> MA 1004: Up to 30. >> Pronod KUMAR BHARATIYA: yeah, if you have to wait for 40 minutes, which is greater than 30 minutes then. when we calculate the probability it will without calculating the probability, we can safely. see that it's probability will be not more than 100 >> MA 1004: Inequality. Everyone. >> Pyari Singh K: Where do we have to use? This? Will we have problems. based on this? >> MA 1004: Yeah, if you have it, given the values of x, which is not negative and the value only mean is you then we can calculate the other one. >> Pyari Singh K: Okay. >> MA 1004: To Mar. the second comes, the geometry So what is Geofficials inequality? It is given as like if x is any random variable, okay? It is not necessary. It should be negative or non-negative, Okay? If x is a random variable and the variables mean, is even that is new is given And the variance is doing, that is Sigma Square. this is the variance for things. It Sigma Square denotes.

obedience and Sigma denotes standard deviation. so, if a random variable x is given, It is a discrete random variable and it's mean and valances. So the trebuchet's inequality is denoted, as probability of mod of X minus 0, Is greater than K Sigma. Is always less than one case. This is the expression of chubby chefs. Here, we have been given the variance and the Marco's inequality only mean was given but in traditions inequality mean and various both of the values are And some constant, right? So like a championship single it also gives the bump right.

It also give us the bound that It states that. That the value of that random variable. Give this the value of that random variable deviates from its mean. It by more than a certain amount. Right. Is it clear? What does it say? It stays that the value of that random variable deviates from its mean by a certain amount. based on its wings right? with some like with some units but, >> Pronod KUMAR BHARATIYA: And this. is variance and it should be smart. >> MA 1004: Sorry. >> Pronod KUMAR BHARATIYA: You are written K into Sigma.

So it is an m signals, a standard deviation. >> MA 1004: Yes. >> Pronod KUMAR BHARATIYA: It will be signed my square. >> MA 1004: No, in this. >> Pronod KUMAR BHARATIYA: but it to be very >> MA 1004: We are taking the sigma that is >> Pronod KUMAR BHARATIYA: To sing. >> MA 1004: static equation. >> Pronod KUMAR BHARATIYA: So k times the standard deviation. >> MA 1004: We are k times the standard >> Pronod KUMAR BHARATIYA: Standardization. >> MA 1004: deviation. >> Pronod KUMAR BHARATIYA: Okay. >> MA 1004: Okay. so for example, like let's take an example. so, the average score or in an exam like 7.

Okay, the expectation is 70 and we have been given the variance as hundred. So, if we have to calculate the probability that a random click randomly selected score, they read from the mean, Okay, if x is a variable, right? So x is getting deviated from the mean. Okay? This getting debuted from the mean by more than 20 points. Okay. So, we have to calculate the probability. this probability, we have to calculate, right? So if you just compare this equation and this equation, right? so here 20 is equal to >> Pyari Singh K: 1 by 5. No. >> MA 1004: Basic it. You getting my point. >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes, ma'am. >> Pyari Singh K: K equal to 2. >> MA 1004: Yeah. so here the bar if you have to calculate the value of k, K will be equal to 20 upon Sigma, right? here the value of Sigma is 10.

>> Pyari Singh K: 10. >> MA 1004: which will be good. K is equal to. so it's minus u greater than 20 will be will be less than equal to 1 upon case square. and value of case 1 by 2 square. that is 1 by 4. Right. And the one by four is equal to 0.25, something, right? So that means that when like at most 25% of the students scored more than 20 points from the meat. Getting you getting my point. Everyone. Sure, everyone. >> 23F3004210 CAESAR PARTHO KARMOKAR: Yes, ma'am.

>> MA 1004: Okay. so when do we calculate the markups inequality? >> Pyari Singh K: And we have only the mean expected >> MA 1004: Yeah. >> Pyari Singh K: value. >> MA 1004: Okay. and more expected. What is the condition like? One more condition was there. >> Pyari Singh K: Greater than a constant. >> P Baskaran Nadar: Values. >> MA 1004: Law negative understanding. And when do we calculate the turbitions inequality? >> P Baskaran Nadar: For all.

>> MA 1004: Like, yeah. for all when we are >> Pyari Singh K: oh so in the case of okay, so in the case of chebyshaves, it need not be >> MA 1004: Okay. >> Pyari Singh K: Values alone. >> MA 1004: Yeah. it should be a discrete randomly. >> Pyari Singh K: Okay. >> MA 1004: This, there is one another form of turbitious inequality in this. We got our upper ball, okay? so if you have to calculate the lower bound we can calculate it as probability of Minus T Sigma.

President. Listen pitch. What's this? Will give us the lower boundaries. So, if the question is, it has been hours about the upper one, then we will apply this. formula if it is as mean as above the Google. Apply this. Is it clear? Find by everyone. No doubt. >> P Baskaran Nadar: This is a lower bound. What does it signify? It's if the complement >> MA 1004: because, >> P Baskaran Nadar: of the other one, >> MA 1004: Yeah. What just see. No. Like, peace probability is greater than this value. So >> P Baskaran Nadar: no, if the probability of it falling within that range beyond the range is one by k squared for falling within this one. Anyway, complement of it. Why do we use the term lower bound? >> MA 1004: like if it's in the question, it has been asked about you what? The plan? be. Like that. there are some questions.

>> P Baskaran Nadar: Questions. or frame like this, but the term lower bound gives me misleading concept. You know, you have something to call an upper bound. lower bound of a probabilities. What? >> Rahul: Decide should be the same exact same and then >> P Baskaran Nadar: It's no different. It's too formulas. One of the same. >> Rahul: Yes, yes. >> MA 1004: yeah, it's same movie but if it has been asked about the Like in. >> P Baskaran Nadar: I'm trying to understand the >> MA 1004: like we can say, >> P Baskaran Nadar: terminology used in this course. Upper one makes sense. Okay. But when we say lower bound, something, which is varying. We are fixing an upper bound understandable. if you are fixing a lower bound, lower bound for a probabilities. 0. Why are we using the word lower? >> MA 1004: Start necessary that always the lower bound is you Like, it can be like the probability can be greater than equal to.

You can say, one by three or it can be greater than two by five. something like that. Like, the probability starts from this value. It is not necessary that all the lower bound is starts from 0. Getting my point. Hello. >> P Baskaran Nadar: I'm thinking, I'm able. >> MA 1004: Yeah. >> P Baskaran Nadar: to figure out why we use the technology because I see no. No different between the two formulas. >> MA 1004: Yeah, there is no difference in this, We are just calculating the complement. >> P Baskaran Nadar: Yeah. >> MA 1004: Yeah. >> P Baskaran Nadar: so it's a complement of that. >> MA 1004: this is this just this is just a compliment. Okay, there is like like if it's the question is given that? what is the probability? that it takes value between? Just say 30 or 50, something like that.

Then we calculate this one. What is the lower bound? That the x takes values between 30 and 50. >> P Baskaran Nadar: That's okay. >> MA 1004: Then we >> P Baskaran Nadar: That's all. But, why are using the word? Lower bound. That's what. Upper bound mix. Meaningful. Yes, it cannot be more than that. >> MA 1004: Yeah. >> P Baskaran Nadar: That is why we call it as an upper bound. That it lattice, it conveys a meaning or a concept. Wonder what conditions we say that we are guaranteeing it that it is not going to be less than this. You get my point. >> MA 1004: Stating that it starts like the probabilities. >> P Baskaran Nadar: What we're saying is the probability of falling within this range given is greater than this. That's why we probably say that. If you're saying probability of getting 90% is 25%, the probability of his mark falling within 70 and 50. Was the same.

Sorry, 1004 90 and 50 is greater than 2/3. something like that. Is that the reason you say it is a lower bound. >> MA 1004: Yeah. Yeah. >> P Baskaran Nadar: Okay. Half. Empty half full. >> MA 1004: Like in the dark session, I can clear this out. You're just the question your question is, like why we are saying the lower levels? >> P Baskaran Nadar: Yeah, the term. I mean I'm just concerned about the terms.

We use which are necessarily likely to cause confusion. so that's okay. >> MA 1004: Right. >> P Baskaran Nadar: Earlier we said, What is the probability? it will fall? Beyond the Range. within the range. Now, you are saying it is. Outside. the range, is it? >> MA 1004: Which one? The This is within the range. Only right? The variable the experimental is taking between the two values. So, what is the probability that the >> P Baskaran Nadar: Correct. >> MA 1004: existing? >> P Baskaran Nadar: but the earlier one is within the beyond the range.

>> MA 1004: Beyond that range. Yeah. That more. >> P Baskaran Nadar: That's why I'm saying, it's one of the same. so that the glass is half, empty or half. Fill. It's two different way of looking at the same thing. but then for figure is same. >> MA 1004: Yeah. >> P Baskaran Nadar: One by K's Square. >> MA 1004: One minus 100. >> P Baskaran Nadar: That's okay. It's not a consequent. I mean, important thing to discuss Anything else? you want to discuss today? >> MA 1004: No, no.

>> P Baskaran Nadar: Thank you. >> MA 1004: it's fine. Yeah, this was the last >> P Baskaran Nadar: Thank you. Thank you. very much. >> MA 1004: Yeah. Yeah. so anybody having any other doubts, For today. >> 24F1002378 KHEMRAJ DHANRAJ NAKHATE: Remember. >> MA 1004: Yeah. >> 24F1002378 KHEMRAJ DHANRAJ NAKHATE: Can you explain Activity 2? >> MA 1004: I will explain it on Tuesday. Yeah, Pyari. >> Pyari Singh K: Yeah. it's not related directly to the material. We discussed when we have the solve with the solutions on. means all with this sessions on Tuesdays, right? >> MA 1004: Yeah. >> Pyari Singh K: So, I won't be able to attend it live. but if we have any questions, can we post it on the discord channel discard channel, the discussion forum, Related to the solid with the session. >> MA 1004: Yeah. but oh click on Tuesday.

You can ask forever, sorry. Questions about any doubt who can use it? >> Pyari Singh K: Yeah, I was trying to say that I won't be able to make it live on the live. sessions. So, if we have any adults, can we post it on the forum? Okay. >> MA 1004: Okay, so can we? You always having any doubts? >> Pyari Singh K: Yes, ma'am. Thank you so much.

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