00:01
So in this question, we are given that the pmf of a random variable x is given by p of x is equal to half raised to x.
00:16
For x is equal to 1, 2, 3 till infinity.
00:21
And we are asked to find what is the moment generating function of x.
00:25
What is the mean of x? and what is the variance of x? so the flow chart will be this.
00:32
We can calculate the moment generating function of x using this formula m of t is given by expectation of e raise to t of x e to t x sorry and after having done that so here next what we'll do is we can calculate the mean of x using this m prime at 0 of this moment generating function m prime is differentiation with respect to t this will give us the mean of x and from this also we get that m double prime at zero will give us expectation of x squared so these are known formulas given in the textbook so using this we can find expectation of x squared and after this we can calculate variance of x as expectation of x random variable x.
01:40
So this is given by expectation of e.
01:43
So e.
01:45
X is a continuous function of x, 1 1 continuous function.
01:49
So we can use the formula for calculating the expectation.
01:55
So e.
01:56
2x, p of x and summation over all x belonging to support of x.
02:03
So this will be e.
02:04
X times p of 1.
02:06
So replacing 1 in place of x plus e.
02:10
To 2 t p of 2 and so forth.
02:17
Now probability of 1 is given over here, we replace x by 1.
02:26
So e.
02:27
2 times half plus e.
02:28
2 t times half raise to 2, plus e.
02:33
3 t times 1 by 2 raised 2, raised 2, and so forth.
02:40
So notice that this series is a geometric progression and the ratio is 1 by 2 is common in each step.
02:52
So we are multiplying the previous term by 1 by 2 times e -race to t.
02:58
So we are multiplying e -r -rish to t times half for each of the previous terms.
03:06
So the infinite sum of a gp is given by a by 1 minus r where a is the 4.
03:13
First term and r is the common ratio.
03:20
So we replace the first term is e to t times half...