STAT 609 First Exam 11:00am-12:00noon, Oct. 3, 2014

Please show all your work for full credits.

1. Let A, B, C, and D be four events. Suppose that if all B, C, and D occur, then A occurs. Show that P (A) ≥ P (B) + P (C) + P (D) − 2.

2. A red die and a white die are rolled, where each die has 6 faces numbered 1, ..., 6.

(a) Are the events A = {4 on the red die} and B = {sum of dice is odd} independent?

(b) Are the events C = {5 on the red die} and D = {sum of dice is 11} independent?

(c) Calculate the conditional probability that the sum of dice is larger or equal to 11, given that 5 on the red die.

3. Let X be a random variable having pdf

f(x) = \{ \begin{matrix}x & 0 < x ≤ 1 \\ 2 − x & 1 < x ≤ 2 \\ 0 & otherwise \end{matrix}

(a) Find the cdf of X.

(b) Obtain E(X).

(c) If Y = X^{2}, find the pdf of Y.

(d) Find E(Y), and use the result to find Var(X).

4. Let X be a discrete random variable with P (X = a) = p and P (X = b) = 1 − p, where 0 < p < 1 and a < b. Show that

lim_{n→∞} \frac{E(X^{n+1})}{E(X^{n})} = b

5. Let X be a discrete random variable with pmf

f_{p}(x) = \frac{1}{log(1/p)}x^{−1}(1 − p)^{x}, x = 1, 2, ...

where 0 < p < 1.

(a) Obtain E(X) by directly calculating

E(X) = ∑_{x=1}^{∞} xf_{p}(x)

(b) Using the fact that f_{p}(x) is a pmf for any p ∈ (0, 1), show that the mgf of X is

M_{X}(t) = \frac{log(1 − (1 − p)e^{t})}{log(p)}, t < − log(1 − p)

(c) Find E(X) by differentiating M_{X}(t) and show the result is the same as that in part (a).