1. Suppose that X is uniformly distributed on (0,3), Y is uniformly distributed on
(0,5), and X and Y are independent. Determine the distribution of the sum X + Y.
2. The moment generating function M X (t ) for a random variable X is
⎛ 2 ⎞
M X (t ) = ⎜
⎟
⎝ 2 − 3t ⎠
3
(a) (8 points) Identify the distribution and the parameter(s) of the distribution.
(b) (6 points) Determine the mean and variance of X.
(c) (6 points) Determine Pr ( X > 3).
3. A random variable X, representing the length of time, in minutes a particular
credit card payment system is down at the local grocer, has a cumulative
distribution function given by
⎧ 0
⎪
FX ( x ) = ⎨ ⎛ 2 ⎞2
⎪1 − ⎜ x ⎟
⎩ ⎝ ⎠
x<2
x≥2
If the cost (in lost sales) for the grocery store is Y = 3 X 2 dollars, determine the
expected loss the grocery store will suffer the next time the system goes down.
4. Suppose that X is uniformly distributed on (0,3), Y is uniformly distributed on
(0,5), and X and Y are independent. Determine the probability that their product
in less than 5.
et
A distribution has moment generating function M X (t ) =
, where t < 1.
5.
1− t
(a) Determine the mean of this distribution.
(b) Determine the variance of this distribution.
6. Suppose that X is a discrete random variable with probability mass function
⎧1
⎪ 6 if x = −1
⎪
⎪1
p X ( x) = ⎨
if x = 0
⎪3
⎪1
⎪ 2 if x = 2
⎩
(a) (12 points) Determine the moment generating function for X
(b) (8 points) Calculate
E ⎡⎣ X 2 − X ⎤⎦
.
⎛1
⎞
Pr ⎜ ≤ X ≤ 1⎟
(c) Determine ⎝ 2
⎠
7. One insurance clerk works on auto policies, another writes up homeowner
policies.
On a random day, let X be the fraction of working day during which the first
clerk writes up on auto policies, and Y be the fraction of day spent on homeowner
policies by the second clerk.
Suppose the joint density of
f X ,Y ( x, y ) =
( X ,Y )
is given by
6
x + y 2 ) , 0 ≤ x ≤ 1, 0 ≤ y ≤ 1.
(
5
(a) Find the probability that both spend less than half the day writing policies.
(b) Find the marginal density function f X of X
(c) Find the probability that first clerk writes up auto policies for at least one
third of the day.
8. The probability mass function for a particular discrete random variable X having
nonnegative integer values is defined by the recurrence relation
3
p X ( x − 1), for x = 1, 2,3,...
5
Determine the probability that X equals zero. Determine the probability that X is
less than 3.2.
p X ( x) =
9. Marie is getting married tomorrow, at an outdoor ceremony in the desert. In
recent years, it has rained only 5 days each year, so the probability it will rain
tomorrow is 5/365. When it rains, the weatherman had correctly forecasted rain
90% of the time. When it doesn't rain, he forecasted that it would rain 20% of the
time.
(a) What is the probability that the weatherman will forecast rain tomorrow?
(b) If the weatherman has predicted rain for tomorrow, what is the probability that
it will rain?
10. Deer ticks can be carriers of either Lyme disease or human granulocytic
ehrlichiosis (HGE). Based on a recent study, suppose that 20% of all ticks in a
certain location carry Lyme disease, 10% carry HGE, and 20% of the ticks that
carry at least one of these diseases carry both of them. If a randomly selected tick
is found to have carried HGE, what is the probability that the selected tick is also
a carrier of Lyme disease? What percentage of ticks carry neither disease? What
percentage carry both?
11. A game is proposed in which the participant spins the wheel below. If the result
is black you lose, in dollars, the amount indicated on the wheel. If the result is
white, you win the amount, in dollars, indicated on the wheel.
(a) Determine the probability mass function for the random variable X that
represents the payout of the spin.
(b) Determine the expected payout.
(c) Suppose you will be charged $0.25 to spin the wheel. Should you play the
game?
Bonus (10 points) The lifetime, in years, of a particular condoogle (a race of
extraterrestrial whose 40th birthday celebration involves a self-termination ceremony) is a
mixed random variable on the interval (0,40] with continuous density function given by
fX ( x) =
k
(10 + x )
2
, 0 < x < 40
(a) If only 5% of condoogles survive to participate in their 40th birthday
ceremony, determine the value of k.
(b) What is the expected lifespan of the condoogle?
(c) Determine the probability that a particular condoogle perishes before its 10th
birthday.