Math 180B
(P. Fitzsimmons)
Practice Final Exam
March 15, 2013
1. Let X and Y be standard normal random variables, with correlation = .4.
(a) The random variable W = 2X + 3Y also has the normal distribution. Find E(W )
and Var(W ).
(b) For any real number
Math 180B
Practice Midterm Exam
January 29, 2013
1. Consider a Markov chain with state space cfw_0, 1, 2 and transition matrix
0
P = 1/3
1/3
1/2 1/2
0
2/ 3 .
1/3 1/3
Suppose P (X0 = i) = 1/3 for i = 0, 1, 2.
(a) Compute P (X2 = i|X0 = 0) for i = 0, 1, 2.
Math 180B, Winter 2013
Homework 2
6.3.2. Let X and Y have the following joint density:
f (x, y ) =
2x + 2y 4xy, for 0 x 1 and 0 y 1;
0,
otherwise.
(a) Find the marginal densities of X and Y .
(b) Find fY (y |X = 1/4).
(c) Find E(Y |X = 1/4).
6.3.5. Suppos
Math 180B, Winter 2013
Homework 3
6.5.4. Suppose X and Y are standard normal variables. Find an expression for P(X + 2Y 3) in terms
of the standard normal distribution function ,
(a) in case X and Y are independent;
(b) in case X and Y have bivariate norm
Math 180B, Winter 2013
Homework 1, due January 15
5.2.16.
Suppose X1 , X2 , X3 are independent exponential random variables with parameters
1 , 2 , 3 respectively. Evaluate P(X1 < X2 < X3 ).
5.3.2. Let X and Y be independent random variables, with E(X ) =