theory of probability 2

MATH 464

HOMEWORK 4

SPRING 2013

The following assignment is to be turned in on Thursday, February 14, 2013.

1. Let X be a discrete random variable on a probability space (Ω,F,P). Let g : R → R be a function and set Y = g(X), i.e. Y : Ω → R is defined by

Y (ω) = g(X(ω)) for all ω ∈ Ω .

Prove that Y is a discrete random variable.

2. Let 0 < p ≤ 1 and consider a function X with range {1, 2, 3, · · ·} and corresponding numbers

P(X = k) = p(1 −p)k−1 for any integer k ≥ 1 .

Prove that X is a discrete random variable by showing that the sum of the above probabilities is 1. This is the geometric random variable with parameter p.

3. Let X be a Poisson random variable with parameter λ > 0. Compute the following:

a) P(2 ≤ X ≤ 4) b) P(X ≥ 5) c) P(X is even)

give each answer in exact form and, with the choice of λ = 2, give a decimal approximation to the above which is accurate to 3 decimal places.

4. Let X be a discrete random variable whose range is {0, 1, 2, 3, · · ·}. Prove that

E(X) = ∞∑ k=0

P(X > k) .

5. Compute the expected value of the geometric random variable with pa- rameter 0 < p ≤ 1. Hint: Use problem 4 above.

6. Let X be a binomial random variable with parameters 0 ≤ p ≤ 1 and n > 0 an integer. For any 0 ≤ k ≤ n, denote by Pk = P(X = k). Compute

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2 SPRING 2013

the ratio Pk−1 Pk

for 1 ≤ k ≤ n.

Show that this ratio is less than one if and only if k < np + p. This shows that the most probable values of X are those near np.

7. Let X be a Poisson random variable with parameter λ > 0. Let g : R → R be the function g(x) = x(x− 1). Set Y = g(X). Find E(Y ).

8. Let X be a function whose range is {1, 2, 3, · · ·}. Consider the values

P(X = n) = 1

n(n + 1) for any n ≥ 1 .

Does this function X define a discrete random variable? If so, what is E(X)?