Generating Functions

Question 1

Define a generating function.

Answer 1

Question 2

Define the convolution.

Answer 2

Question 3

What is the generating function of c when a and b have generating function G_a(s) and G_b(s)?

Answer 3

Question 4

What are four important properties of generating functions?

Answer 4

Question 5

Define a probability generating function.

Answer 5

Question 6

Finish the following theorem.

Answer 6

Question 7

Prove the following theorem.

Answer 7

Question 8

How do you compute the variance through a moment generating function?

Answer 8

Question 9

How do you derive the moment generating function using the probability generating function?

Answer 9

Question 10

Finish the following theorem about generating functions.

Answer 10

Question 11

Prove the following theorem.

Answer 11

Question 12

What is the left and right hand side of the following called?

Answer 12

The left hand side is a convolution of generating functions while the right side is a multiplication of generating functions.

Question 13

Finish the following theorem.

Answer 13

Question 14

Prove the following theorem.

Answer 14