Exercise 8.5. Find the mean and variance of the random sum by the help of the Laplace-transform.

Solution:

Therefore

Exercise 8.6. Find the Laplace-transform of the Erlang distribution with parameters and then the mean and variance.

Solution:

Therefore

Exercise 8.7. Find the mean of the hyperexponential distribution by the help of the Laplace-transform.

Solution:

Exercise 8.8. Find the Laplace-transform of the hypoexponential distribution.

Solution:

By applying the properties of the Laplace-transform we have

Exercise 8.9. Find the Laplace-transform of the Gamma distribution.

Solution:

where .

Exercise 8.10. Show that if , and are independent random variables, then

Solution:

which proves the statement.