1. (6 pts) The lifetime of a computer costing $900 is exponentially distributed with mean 3 years
The manufacturer agrees to pay a full refund to a buyer if the computer fails during the first
18-month following its purchase, a one-half refund if it fails during the second 18-month, and
no refund for failure after the third year. Find the expected total amount of refunds from the
sale of 300 computers.
2. (5 pts) A is waiting for bus Q85 in Jamaica Station. The probability that the next bus is Q85
is 0.2. If the first three buses are not Q85, what is the probability that Q85 will come after the
first 5 buses?
3. (6 pts) On one street, a drivers speed just before an accident is uniformly distributed on [15, 25].
Given the speed, the resulting loss from the accident is exponentially distributed with mean of
1.25 times the speed. Calculate the variance of a loss due to an accident on that street.
4. (12 pts) Let the joint p.m.f. of X and Y be defined by
f(x, y) = 3x + y
45
, x = 1, 2, 3 y = 1, 2
a) Find fX(x), the marginal p.m.f. of X, and fY (y), the marginal p.m.f. of Y .
b) Are X and Y independent? Please explain your reasoning.
c) Find P(X + Y < 4).
d) Find E(XY ).
5. (9 pts) The life of a certain automobile tire is normally distributed with mean 60,000 miles and
standard deviation 5000 miles.
a) What is the probability that such a tire lasts less than 54,000 miles?
b) Given that is has survived 65,000 miles, what is the conditional probability that the tire
survives another 7,000 miles?
c) If a chosen tire is on the middle 75% of life, what is the minimum miles that it can last?
6. (12 pts) The probability density function of a random variable X is
f(x) =
2.5 −k <= x < 0
2e
−4x 0 <= x < infty
a) Determine k.
b) Find the cumulative density function F(x).
c) What is the median pi 0.5?
d) Find P(X > 5|X >= 2).