theory of probability questions.

MATH 464

HOMEWORK 5

SPRING 2013

The following assignment is to be turned in on Thursday, February 21, 2013.

1. Let x,y ∈ R and take n ≥ 2 an integer. Prove that

(x + y)n =

n∑ k=0

( n

k

) xkyn−k

using an induction argument.

2. Let X be a binomial random variable with parameters n and p. Find the mean and variance of X. Hint: Sometimes it is easier to calculate E(X(X − 1)) = E(X2) −E(X) rather than E(X2) directly.

3. Let X be a geometric random variable with parameter p > 0. Find the mean and variance of X. Hint: By geometric series, we know that

∞∑ n=0

rn = 1

1 −r for any real number r with |r| < 1.

By taking derivatives, you can get some useful formulas: ∞∑ n=1

nrn−1 = d

dr

1

1 −r and

∞∑ n=2

n(n− 1)rn−2 = d2

dr2 1

1 −r

4. a) Let X be a discrete random variable. Show that if E(X2) = 0, then P(X = 0) = 1.

b) Use part a) to prove that if X is a discrete random variable and var(X) = 0, then P(X = µ) = 1 where µ = E(X).

5. Let X be a geometric random variable with parameter p > 0. Let m and n be non-negative integers. Show that

P(X > n + m |X > m) = P(X > n) . This shows that X has the lack of memory property.

6. Let X be the number of eggs laid by an insect. Suppose that X is a Poisson random variable with parameter λ > 0. Suppose that each egg produces an insect with probability 0 < p < 1, and assume that the eggs

1

2 SPRING 2013

are independent of each other. Let Y be the number of insects that hatch from the X eggs.

a) Find E(Y ).

b) Show that Y is also a Poisson random variable by calculating its param- eter (in terms of λ and p).

Hint: For both parts above, use the partition theorem with Bk = {ω ∈ Ω : X(ω) = k} with k ≥ 0. Note also: If we are given that X = k, then Y is a binomial random variable.