Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

MATH20712: Random Models

Problems Week 2

1.     (i) Write down the definition of the probability generating function of a random variable X.

(ii) Assume that X has probability mass function

P(X = 0) = 1 , P(X = 1) = 1 , P(X = 3) = 1 , P(X = 4) = 1 , P(X = 5) = 1

Compute the probability generating function of X.

2.     (i) Assume that X has the probability generating function: Gx(s) = + s + s2 + s3 + s7 .

Write down the probability mass function of X.

(ii) Assume that X has the probability generating function

GX(s) = ╱ + s、10 .

Write dow the probability mass function of X. What is the distribu- tion of X?

3. Assume that X has the probability generating function GX(s) = e ─5+5s . Write down the probability mass function of X.

4. X has a Geometric distribution:

P(X = k) = q k ─1p, k = 1,2, ... ; 0 < p < 1, p + q = 1.

(i)  Find the probability generating function of X and E(X).

(ii)  If X1 , X2 are independent and have the same distribution as X, wirte down the probability generating function of S = X1+ X2 .

5.     (i) X has the Poisson distribution with parameter λ, i.e.

λn

n!

Find the probability generating function of X and 5X + 2.

(ii) X has the porbability generating function GX(s) = .  Find the

probability mass function of X and write down the probability gen- erating function of 3X + 5.

6.  Let X1  ~ Bern(p1), X2  ~ Bern(p2) and X3  ~ Bern(p3) be three indepen- dent Bernoulli random variables with parameters 0 < p 1 , p2 , p3 < 1. Find the probability mass function of Y = X1+ X2+ X3 .

7.  Let X be a discrete random variable with probability generating function GX(s) = (1 ─ p 1+ p 1 s)n . . eλs─λ ,

where 0 < p 1 , p2 < 1, n 2 1 and λ > 0. Find E(X) and P(X = 1). Further- more, write down the probability mass function of X.

8.  (*) Let X ~ Geo(p) be a geometric random variable with parameter p . Let Y ~ Bin(X, p_), which means that, conditioned on the event X = k, Y is a binomial random variable with parameter (k, p_). Find E( Y).

9.  (*) Let X ~ Bern(p1) and Y ~ Geo(p2) be two independent random vari- able. Find the probability generating function and the probability mass function of Z = X x Y .