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University of Wisconsin Milwaukee Department of Economics Fall 2004 Ozlem Eren
ECON 413 MIDTERM 2
- The joint distribution of the random variables X and Y is given in the following table. Y
0 1 2
0 0.1 0.2 0.
X 1 0.1 0 0.
2 0 0 0.
a) Find E(X⏐Y = 1)
b) Find Var(X) and Var(Y)
c) Find ρX,Y.
d) Find Var(X-Y)
- A discrete random variable X has the following cumulative distribution function
for x < 0
F ( x )
for 0≤ x <
for 1≤ x <
for 3≤ x <
for x ≥ 5
a) Find the probability distribution of X, i.e find f(x), x = 0,1,…,
b) Find P(X = 3)
c) Find P(X ≤ 3)
University of Wisconsin Milwaukee Department of Economics Fall 2004 Ozlem Eren
d) Find P(X ≥ 1)
e) Find P(1 < X < 5)
- Suppose accidents occur at a manufacturing plant according to a Poisson process at the rate of 2 accidents per month.
a) What is the probability that no accident will occur over a one-month period?
b) What is the probability that more than two accidents will occur over a one- month period?
c) What is the probability that no accident will occur over a three-month period?
d) What is the probability that more than two accidents will occur over a three-month period?
- Answer the following questions. Show all your work, otherwise you will not get any credit.
a) Let X 1 and X 2 be two random variables with means μ 1 and μ 2 and variances σ 12 and^ σ 22 , respectively. Let Y = k 1 X 1 + k 2 X 2 where k 1 and k 2 are fixed constants. Find E(Y) and Var(Y).
b) Let X, Y, and Z be uncorrelated random variables. Define U = Z + X and V = Z + Y. Find Cov (U,V)
- Four men whose coats appear identical hang them up in a coat room and later pick them up at random. Let X denote the number of matches (i.e., the number of men who get their own coats.) Define the indicator variable Yi, which is 1 if Mr. I gets his own coat and 0 if he does not.
a) Find P(Y 1 =1).
b) Find EY 1
c) Note that the Yi’s are exchangeable, and use this fact to find EX.
d) Show that if there are n men and n coats, the expected number of men who get their own coats is the same for all n.
