 Dear students, let me present to you the concept of moments. As you already know, this is one of the very, very fundamental and important concepts in whenever we are talking about random variables and probability distributions of random variables. So let us do it step by step. Subsequently, let us define the emeth moment about an arbitrary origin. Consider the expression, expected value of x minus a raised to m, where m is a positive integer and a is any real number, arbitrary real number. All right, if we consider this expression, what is the detailed formula of this expression? Well, of course, if we are dealing with a continuous variable, this expectation will be given by the integral from minus infinity to infinity of x minus a whole raised to m multiplied by f of x. And on the other hand, if we are talking about a discrete variable, then this expectation is given by the summation of x minus a raised to m multiplied by p of x. That is, these are the same things. You know that whenever there is a continuous variable, then the integration is done. And when there is a discrete variable, then the summation is done. Now, the thing to note in this is that whatever I have put in front of you, this is the formula of the emeth moment about an arbitrary origin. Just as I said, m is a positive integer. So m can be equal to 1 or 2 or 3 or 4 and so on. So we say first moment, second moment, third moment, fourth moment, and so on. But what we have to note is that this is an arbitrary origin, that is, a is an arbitrary real number. Please also pay attention to its notation. This kind of moment is denoted by mu dash m. That is, we will write m in the subscript and on top of it, we will write a dash or a prime. We usually call it dash em. Mu dash m. This is an arbitrary origin, so this is the notation. Now, after a while, you will see that when we talk about the mean, at that time, this dash is not attached to it. So that it can be differentiated between that and this. Okay, just like I said, m is a positive real number. So here as well, the mu dash m, if it is m1, so we are talking about mu1 dash, if it is m2, so mu2 dash, mu3 dash is just right. All right, now, let us talk now about the emeth moment about the mean. The expression you just gave, if we put a equal to mu, that is, a is an arbitrary number. So it can also be equal to mu. It can be any number, so it can be equal to the mean. It can be equal to mu. So if we put mean or mu, then what will be the expression? It will become expected value of x minus mu raised to m. So this is the emeth moment about the mean. Now, you are noting that the things subtracting from x, we say it's the moment about that. Again, if it is a continuous variable, we have this thing is equal to minus infinity to infinity, x minus mu raised to m into f of x, that is, the integral that we took. And if it is discrete, it is the sum of x minus mu raised to m multiplied by p of x. We will sum up all the x values of that kind of products. Its notation, as I just said a little while ago, is in front of you. In this one, dash is not attached to it. So if we put m to 1, 2, 3, 4, then we will have mu 1, mu 2, mu 3, mu 4, and so on. And the expression is in front of you. Okay, note that the moment about the mean, these are also called central moments. Last but not the least, let me talk about the emeth moment about the origin. Now, you will say, what is this? I was saying this a little while ago. No, previously I said arbitrary origin. Now, I am saying the origin. And note that if I say only origin, then I am talking about zero. So I am talking about zero. Because on the graph paper, which we start making graph from a school level, do you agree or not? Where zero is the intersection of y-axis and x-axis, which is taken at zero, is that not called the origin? So according to that, note that if you are saying arbitrary origin, then we are talking about A. But if we are saying only origin, then actually we are talking about zero. So if we put that arbitrary number A equal to zero, what do we get? Expected value of x minus zero whole raised to M. So it is equal to expected value of x raised to M. So this is it. The expected value of x raised to M is the mth moment about zero. Yeah, mth moment about the origin. And again, if it is a continuous variable, this thing will be equal to the integral from minus infinity to infinity of x raised to M into f of x. But if it is a discrete variable, it will be the sum of all such products, x raised to M into P of x, as many x values we may have. Iski notation peh gaur ki jhe, jo pehle notation main aapke saamne rakhi thi for arbitrary origin, yani wo dash dis ke andar tha, iske liye bhi students, wohi notation isthimal ki jaati. So mu dash m equal to e raised, magar woh context se pata lag jaata ek woh hai ki ye hai. So agar ye hai, so then mu dash m ya mu m dash is equal to e of x raised to M. Ab iske baad mai aapke saamne do special case rakhna jaati hoon. Pela special case dekhye ki ye jo abhi abhi aapke saamne prezent kiya hai, mth moment about the origin. Iski andar, if we put m equal to one, what do we get? We get instead of mu m dash, of course we will now have mu one dash and it is equal to what? Expected value of x raised to M, m ko one rakh diya to kya aaya? Expected value of x raised to one, yani expected value of x. So what is expected value of x? Of course it is the mean. So this is the first special case. K, the first moment, kyu ka m ko one rakh hai na? So the first moment about the origin is simply the mean of the distribution. Iske baad, second special case dekhye aur ye wala jo formula mai aapko central moments ka diya hai, yani moments about the mean ka jo zikar kiya hai abhi thori der pehle uswale ke saath us ka link hai aur agar hum uske andar m ko two rakhte, to kya ho ga? mu m is equal to something hum ne kaha tha tab, to ab hum m ko two kare hain. So mu two, mu two, the second moment about the mean ya du se lafzo mai, the second central moment is equal to, according to the formula that I just presented, expected value of x minus mu whole raise to m, they can expect m is equal to two. So expected value of x minus mu whole square. So what is expected value of x minus mu square? Students, it is none other than the variance. It ho variance hota hai na? So therefore this is the second special case. K, the second central moment is none other than the variance of the distribution.