 Dear students, let me present to you the derivation of the moment generating function of the discrete uniform distribution. As you know, a discrete uniform distribution defined on the set A, A plus 1, A plus 2 and so on up to B is denoted by the symbol U, A, B and the constants A and B are known as the parameters of the discrete uniform distribution. MGF, I would like to do it for the simple case, that simple case when A is equal to 1 and B is equal to N. In other words, I am talking about that particular discrete uniform distribution which is defined on the numbers 1, 2, 3, so on up to N. Now in this case, it becomes simple and obvious that because all the various these N possible outcomes are equally likely to occur, then obviously each one of the N probabilities is equal to 1 by N. So given this, now let us try to derive the MGF of this distribution. So what is the basic definition of the MGF? As you know, it is the expected value of e raised to Tx. In the case of any discrete distribution, obviously we will write summation e raised to Tx into P of x, where P of x is the PMF, the probability mass function of that distribution. So let us put it there, what do we get? We get summation e raised to Tx multiplied by 1 over N, so obviously you can take it out. So what do we get now? It is 1 by N multiplied by summation e raised to Tx, but you know that a raised to bc, you can write a raised to b whole raised to c, here I would like to do that. And so what is the expression that I have now? I have the equation mt, mgf of the random variable x is equal to 1 over N summation e raised to T whole raised to x. Students now I would like you to realize that summation e raised to T whole raised to x represents the sum of geometric series, in which the first term A is equal to e raised to T and the common ratio is also equal to A raised to T. Now you will think how can this be, as you can see on the screen those values are going from 1 to N. That is equal to e raised to T whole raised to 1, i.e. e raised to T, it is very easy to verify that the common ratio is also equal to e raised to T. The next point is that we already know from school days that the sum of a geometric series having N terms like this is given by A into 1 minus r raised to N over 1 minus r, where r that common ratio is not equal to 1, so here these values a value e raised to T, r the values e raised to T, substitute it and you get e raised to T multiplied by 1 minus e raised to T whole raised to N over 1 minus e raised to T. So obviously e raised to T whole raised to N can be written as e raised to TN and that gives you that expression. Now let us put it in that initial expression of the MGF. So students, what is the final result? The final result is the MGF M T is equal to e raised to T minus e raised to N plus 1 times T divided by N into 1 minus e raised to T. Obviously you just did a little bit of algebraic manipulation in order to get this final result.