Practice for Final Exam, math 362
1. Let X be a continous random variable with probability density function
fX(x) = 1− |x| −1<x<1;
0,otherwise
Determine the density function for Y=X2.
2. Let X and Y be continuous random variables with joint density function
f(x, y) = 6x0< x < y < 1;
0,otherwise
(A) Find E(X) and E(Y) .
(B) Find Cov(X, Y ).
(C) Find P(X < 1/2, Y > 1/2)
(D) Determine P(X < 1/2|Y= 1).
(E) Determine V[X|Y= 1]
3. Suppose the number of repairs during an 8-hour workday of a machine is a Poisson
random variable with mean 2.
(A) Find the probability that the machine will have 1 breakdown during the day.
(B) Find the probability that the machine will work for at least 4 hours without breaking
down.
(C) Find the density function of the time between starting the machine and the second repair
during an 8-hour workday.
4. An urn contains 5 black balls and 4 white balls. Suppose 4 balls are drawn from the
urn without replacement. Determine the probability of obtaining 1 white ball and 3 black
balls given that at least 2 of the balls drawn are black.
5. Suppose the number of claims, C, from an insurance policy is 0, 1, 2 or 3 each with
probability 1/4. Let L denote the aggregate loss. Suppose that
E(L|C= 0) = 0
V(L|C= 0) = 0
E(L|C= 1) = 5
V(L|C= 1) = 3
E(L|C= 2) = 10
V(L|C= 2) = 7
E(L|C= 3) = 6
V(L|C= 3) = 3
Find V(L).
6. Let the random variable X have the normal distribution with mean µand variance σ2.
Find the density function of Y=eX.
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