Problem 148E

Differentiate the moment-generating function in Exercise 3.147 to find E(Y ) and E(Y 2). Then find V (Y ).

Reference

If Y has a geometric distribution with probability of success p, show that the moment-generating function for Y is

Solution 148E

Step1 of 3:

We have a random variable ‘Y’ and it follows geometric distribution with parameter ‘p’.

Then the probability mass function of geometric distribution is given by:

Where,

x = random variable

p = probability of success(Parameter)

n = sample size

Also we have moment generating function of Y is .

We need to find mean and variance of geometric distribution by using moment generating function.

Step2 of 3:

Consider,

The mean of geometric distribution is given by:

Hence,

Now,

Therefore, The mean of geometric distribution is

Step3 of 3:

Consider,

From step 2, We know that

Now,

Hence,

Now,

Therefore, .

The variance of geometric distribution is given by:

Therefore, The variance of geometric distribution is .