4) Generating Functions

Question 1

What is the probability generating function

Answer 1

Question 2

If X is finite what does it pgf look like

Answer 2

PX(t) is a polynomial function

Question 3

How can you find P(X=k) for a discrete random variable X using its pgf

Answer 3

Question 4

What is the pgf for a Bernoulli distribution

Answer 4

Question 5

What is the pgf for a Binomial distribution

Answer 5

Question 6

What is the pgf for a Poisson distribution

Answer 6

Question 7

What is the pgf for a Geometric distribution

Answer 7

Question 8

How are the expectation and variance of a discrete random variable related to its pgf

Answer 8

Question 9

What is the pgf for the sum of independent discrete random variables

Answer 9

Question 10

What is the moment generating function (mgf)

Answer 10

Question 11

What is the mgf of discrete and continous random variables

Answer 11

Question 12

What are the properties of mgf

Answer 12

Question 13

What is the mgf of the Poisson Distribution

Answer 13

Question 14

What is the mgf of the Exponential Distribution

Answer 14

Question 15

What is the mgf of the Standard Normal Distribution

Answer 15

Question 16

What is the mgf of the Normal Distribution

Answer 16

Question 17

What is the k-th moment of a random variable

Answer 17

Question 18

What are the properties of the mgf if MX(t) < ∞ for all t such that |t| < δ, for some δ > 0

Answer 18

Question 19

What is the mgf of the sum of independent random variables

Answer 19

Question 20

What is the gamma function

Answer 20

Question 21

What are the properties of the gamma function

Answer 21

Question 22

What is the pdf of the gamma distribution

Answer 22

Question 23

What is the mgf of the Gamma Distribution

Answer 23

Question 24

What is Wald’s identity

Answer 24

Question 25

What are the properties of the mgf and pgf for a sum of i.i.d. random variables

Answer 25