 Dear students, I would like to present to you the concept of transformation in the case of a bivariate probability mass function. Let X1, X2 be a random vector and let us suppose that we know the joint distribution of X1, X2 and we seek the distribution of a transformation of X1, X2. We want to say Y is equal to GE of X1, X2. For example, let Y be equal to X1 plus X2 and now we are interested in finding the distribution of Y. Ex-tension is that you may be wanting to define two new variables Y1 equal to G1 of X1, X2 and Y2 equal to G2 of X1, X2. And if you are doing this, now you are interested in finding the joint distribution of Y1 and Y2. So the question is that how can we deal with it, how can we tackle it. So if I concentrate on the discrete case which I am talking about at this point in time, then I can proceed as follows. If we let P of the random variables X1, X2 at the point small x1, small x2 represent the joint PMF, the joint probability mass function of two discrete random variables X1 and X2 with the set of points which is denoted by S which is the support of X1, X2 being of course that set of points where these probabilities are positive greater than 0. Given this students let us define the random variable Y1 as the function U1 of X1, X2 and the random variable Y2 as the function U2 of X1, X2. Obviously these are just notations and you can use any convenient notation. Now if I define this Y1 and Y2 in this way and say that it is a one to one transformation that maps S on to T that is representing the support of the PMF of X1, X2. Similarly T is that set of points that will represent the support of the PMF of these new variables Y1 and Y2. So if we have said that T is the notation that we are going to use for it then the joint PMF of these two new random variables can be written as follows P of Y1, Y2 at the point small Y1, small Y2 is equal to P of X1, X2 at the point W1, Y1, Y2, W2, Y1, Y2 and this is holding for all those ordered pairs Y1, Y2 that belong to T and the PMF is equal to 0 elsewhere. Iskender you must understand very importantly that W1, Y1, Y2 is X1 and W2, Y1, Y2 is X2. Actually these two together form the single valued inverse of Y1 equal to U1, X1, X2 and Y2 equal to U2, X1, X2. This inverse function concept that is very important and this technique this is called the change of variable technique and it needs to be emphasized here that we are having two new variables to replace the two old variables. Now we have new variables Y1 and Y2 and after we have found the joint PMF as I just explained to you then we may obtain the marginal PMF of Y1 by summing over Y2 or the marginal PMF of Y2 by summing over Y1. So this is the concept of transformation in the case of the probability mass function of a random vector consisting of two random variables.