 Dear students, I would like to discuss with you, how we can obtain product moments and simple moments from the moment generating function of a random vector. Let me discuss this with you in the simple case when we have a random vector consisting of only two random variables X and Y. Before I present to you the expressions that are valid in this case, I would like to remind you that you have a procedure for finding the moments in the case of one single variable, right, that if A's variable is AX and its MGF is available to you, then how do you get the moments of that particular random variable or that distribution? Recall that you will put the first derivative of t as 0 and you will get the first moment about 0, then you will get the second derivative and then you will put t equal to 0 and you will get your second moment about 0 and so on. So it is an extension of what we have in the univariate case. So let us now look at this formulae and let us focus on them carefully. Dear students, when we have only one variable, how do we represent the MGF? We write capital M and then we have a bracket and then we have small t, m of t, we are saying and in subscript we can write x and write so that we know that this is for the random variable x. And when we take the derivative later, the derivative is also with respect to t. Now here when we have two, then how will it work? Okay, when we want to find the expected value of x, I will take the partial derivative of the MGF with respect to t1, i.e. we have m of t1, t2 which is available, take its partial derivative with respect to t1, and then substitute t1 equal to 0 and t2 equal to 0 in order to find the expected value of x. In order to find the expected value of y, we will take the partial derivative of the MGF with respect to t2, and after doing that we will put t1 and t2 equal to 0 in order to obtain the expected value of y. Isitara, if I want the expected value of x square, i.e. the second simple moment about 0, then what should I do? I should take the second derivative, partial second derivative of the MGF with respect to t1. To usko ham isitara likte na, curl square by curl t1 square. Or uske ba put t1 and t2 equal to 0 and you will get your e of x square. Similarly, e of y square. Now, let us consider the product moment. If I am interested in obtaining the expected value of xy, i.e. x raised to 1 into y raised to 1, isitara ke moment ko product moment keitein. Ye first hai, ham baadme e of x square into y square bhi leh sakte hain. Or combinations bhi ho sakte hain. Lekin iswak, I will just focus first only on e of xy, expected value of xy, which is in fact a very important product moment. Why? I will tell you in a short while. Lekin pehle ye toh dekhle, ke wo MGF se agar nikal na hai toh kahan se nikalega? Well, it will be curl square by curl t1 curl t2 of the MGF and then put t1 and t2 equal to 0. Yehni jab ham e of x square nikal rehte to curl square by curl of t1 square. Jab ham e of y square nikal rehte to curl square by curl t2 square. Lekin ab jab ham e of x into y nikal rehin, to it is curl square over curl t1 into curl t2. Now that you have obtained these formulae, you can see that you have obtained mu1 and mu2. After all, what is e of x? It is the mean of the random variable x. Similarly, the expected value of y is the mean of the random variable y. But I want you to know that you can also write the expressions of the variance of x and the variance of y. Yehni sigma 1 square and sigma 2 square. Using these expressions that I have just now presented, the derivatives of the MGF and putting t1 and t2 equal to 0. These expressions are now in front of you on the screen. But students, if we have two variables, then we are not interested only in the individual means and the individual variances. We are also interested in the covariance. Aur aap jaanthe hi hai ke covariance ka formula kya hota hai? The covariance is the expected value of x minus mu1 into y minus mu2. And if we express it in another form, which is called the shortcut formula, then this thing is equal to the expected value of xy minus the expected value of x into the expected value of y. So abhi thodi there pehle main aap se kaha ke yeh jo e of xy hai. It is quite important. Aur isski bhaja yeh hi hai ke it is a part of the formula of the covariance. So the mathematical formula that you get in terms of the derivatives of the MGF, if you want to find the covariance. And last but not the least, students, if we have found the covariance and we have also found the variances, then we can very quickly obtain the correlation coefficient because you know that the correlation coefficient is simply the covariance divided by the square root of the variance of x into the variance of y. Correlation ka concept to aap jante hi hai ke bohot important hai. Aur yeh saara kuch jo aap ne kiya. Uske zeh rye you can find the correlation coefficient using the MGF of the joint distribution if this MGF is easily available. Baaz okaath aap jante hi hai ke MGF does not exist. Uske iss me to yeh saari baat valid nahi hai. But if the MGF is there or if you can derive it very easily, then you can use it in this manner to find the means, the variances, the covariance and the correlation coefficient.