Download Math 361 sample final exam

Math 361 Sample Final Exam
Show all your work to qualify for full credit.
Calculators, books and notes are not allowed on this test.
1. Suppose that A1 , A2 and A3 are independent events with
1
1
1
P (A1 ) = , P (A2 ) = , P (A3 ) = .
2
3
4
Find P (A1 ∪ (A2 ∩ A3 )).
2. The probability that a randomly chosen male has a blood circulation problem is 0.25.
Males who have a circulation problem are twice as likely to be smokers as those who
do not have a circulation problem. What is the conditional probability that a male
has a circulation problem, given that he is a smoker?
3. Let X be a random variable with cumulative distribution function


if x < 1,
0
1
2
FX (x) = 2 (x − 2x + 2) if 1 ≤ x < 2


1
otherwise.
(a) Compute: (i) P (X = 1),
(ii) P (X = 2)
and (iii) P (X > 2).
(b) Find fX (x), the probability density function of X. Check that your f (x) is indeed
a p.d.f.
4. A certain flight uses a plane which has 40 seats, and 45 tickets have been sold (overbooking). Assume that each passenger will show up for this flight with probability
0.95, independent of other passengers. Find the probability that all ticketed passengers
showing up for this flight can be accommodated.
5. Suppose that X is an exponential random variable with parameter λ = 2, Y is a
geometric random variable with parameter p = 1/3 and that X and Y are independent.
Find
(a) E[4X − 2Y + 1].
(b) Cov(2X + Y, X + 3Y ).
6. A device runs until either of two components fails, at which point the device stops
running. The joint density function of the lifetimes of the two components, both
measured in hours, is
(
1
(x + y) if 0 < x < 3 and 0 < y < 3,
f (x, y) = 27
0
otherwise.
Calculate the probability that the device fails during its first hour of operation.
1
7. Let X be a random variable with cumulative distribution function
(
1 − ( x2 )2 for x > 2,
FX (x) =
0
otherwise.
Let Y = X 2 . Compute fY (y), the probability density function of Y .
8. The number of people who enter an elevator on the ground floor is a Poisson random
variable with mean λ. There are 10 floors above the ground floor. Assume each person
is equally likely to get off at any one of these 10 floors, independently of where the
others get off. Compute the expected number of stops that the elevator will make
before discharging all of its passengers.
Hint: Conditional expectation, Indicator RV.
9. Let X and Y be continuous random variables with joint probability density function
(
24xy if 0 < x < 1 and 0 < y < 1 − x,
f (x, y) =
0
otherwise.
Compute the conditional probability P (Y < X | X = 31 ).
10. Let X be a discrete Random Variable with probability mass function given by
x
1
2
, x = 1, 2, . . .
f (x) =
2
3
(a) Compute the moment generating function of X, MX (t) (Do not leave your answer
in the form of an infinite sum!).
(b) Find E[X].
11. Suppose that X and Y are independent RVs. Given that X is normal with parameters
2
µX = 2 and σX
= 1, Y is normal with parameters µY = 1 and σY2 = 3. Find the
probability P (X > Y ).
12. Suppose that U and V are independent exponential RVs with parameter λ = 1
(a) Find the probability density function of X = min(U, V ).
(b) Find E[X].
13. Let X1 , . . . X100 be independent, identically distributed random variables with mean
µ = 2 and variance σ 2 = 4. Use the central limit theorem to find an approximation of
the probability
!
100
X
P
Xi > 220 .
i=1
2