STAT 609 Second Exam 11:00am-12:00noon, Nov. 7, 2014

Please show all your work for full credits.

1. Suppose that the covariance matrix of the 2-dimensional random vector (X, Y) is

( \begin{matrix}1 & 1 \\ 1 & 4 \end{matrix} )

(a) (10 points) Find the covariance matrix of (X − 2Y, 2X − Y).

(b) (10 points) If the joint distribution of (X, Y) is normal, show that it is from an exponential family with the form

f_{θ}(x, y) = h(x, y)c(θ) exp (∑_{i=1}^{k} w_{i}(θ)t_{i}(x, y)) ,

and identify k, the value of the parameter vector θ, and functions h, c, and t_{i}’s.

2. Let P be a random variable having pdf

f(p) = \{ \begin{matrix}3p^{2} & 0 < p < 1 \\ 0 & otherwise \end{matrix}

Suppose that given P = p, X and Y are conditionally independent discrete random variables with

P (X = k|P = p) = P (Y = k|P = p) = p(1 − p)^{k}, k = 0, 1, 2, ...

(a) (10 points) Obtain the pmf of X, i.e., P (X = k), k = 0, 1, 2, ...

(b) (10 points) Obtain the pmf of Z = max{X, Y }.

(c) (10 points) Obtain the conditional pdf of P given X = x and Y = y, x, y = 0, 1, 2, ...

3. Let X and Y be independent random variables, X ∼ uniform(0, 1) and Y ∼ uniform(0, 1).

(a) (10 points) Obtain the joint pdf of (X + Y, X − Y).

(b) (10 points) Obtain the conditional pdf of X|(X − Y) = t.

(c) (10 points) Obtain the pdf of |X − Y |.

4. Let X and Y be independent random variables, X ∼ N(0, 1), Y ∼ N(0, 1), and

Z = \{ \begin{matrix}|Y | & X ≥ 0 \\ −|Y | & X < 0 \end{matrix}

(a) (10 points) Show that Z ∼ N(0, 1).

(b) (5 points) Prove that the joint distribution of (X, Z) is not normal.

5. (5 points) Let X and Y be two random variables having mgf’s M_{X}(t) and M_{Y} (s) which are finite for t ∈ (−h, h) and s ∈ (−h, h) with a positive constant h. Suppose that the mgf of the random vector (X, Y) is

M_{(X,Y)}(t, s) = E(e^{tX+sY}) = M_{X}(t)M_{Y} (s), t ∈ (−h, h), s ∈ (−h, h) Show that X and Y are independent.

1(a):

P (Z = 0) = p^{2}

P (Z = x) = 2 ∑_{k=0}^{x−1} p^{2}(1 − p)^{x}(1 − p)^{k} + p^{2}(1 − p)^{2x}, x = 1, 2, ...

2.

( \begin{matrix}13 & 5 \\ 5 & 4 \end{matrix} )

3. The pdf of X − Y is 1 − |t| for |t| < 1.