Exercise 8.1. Find the generating function of the binomial distribution with parameters and then its mean, variance and distribution.

Solution:

Exercise 8.2. Find the generating function of the geometric distribution with parameter p. Furthermore, investigate for which it will be convergent, then calculate the mean and variance.

Solution:

Exercise 8.3. Find the distribution by the help of the generating function

Solution:

Thus

that is, is the generating function of the Poisson distribution with parameter .

Exercise 8.4. Find the mean and variance of the random sum by the help of the generating function.

Solution:

As it has been proved the generating function of the random sum is

Hence

Furthermore

thus

Therefore