# Probability

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MATH 464:

TEST 2

MAKE UP

SPRING 20

1

3

Name

I.D. Number

Question Points Score

1 10

2 10

3 10

4 10

5 10

6 10

7 10

Total 70

1

2 SPRING 2013

Rules to the Make-Up:
Here are the rules to the make-up. You have two choices. Either you

turn in the make-up, or you do not. If you turn in the make-up, it will be
graded, and you will receive a new grade for Exam 2. That new grade will
be the average of your grade on Exam 2 and the Make-Up to Exam 2. If
you do not do the Make-Up, your grade on Exam 2 remains the same. The
Make-Up is due on Tuesday, April 23, 2013.

Do all of the following problems. An answer alone will receive no credit.
Justify all your claims.

(1) You are dealt 5 cards from a standard deck. You keep careful track
of the order of the cards you are dealt.

a) What is the probability that you get one ace and the rest are face
cards? (Face cards are the jack, the queen, and the king.)

b) What is the probability that you have a pair of tens and a pair
of threes (and no better)?

MATH 464: TEST 2 MAKE UP 3

(2) I have 25 brownies and 3 friends.

a) How many ways are there for my friends and I to share these
brownies with no constraints?

b) How many ways are there for us to share the brownies if I insist
that my best friend (one of the three) and I each get at least two
(with no other constraints)?

4 SPRING 2013

(3) Consider an experiment where you roll two fair, 4-sided dice. Label
one as die 1 and one as die 2. Let X be the random variable which
is the sum of the values on die 1 and die 2. Let Y be a random
variable which is the value of die 1 minus the value of die 2.

a) Find the pmfs for X and Y individually. Write them as tables.

b) Find the joint pmf of X and Y . Write it as a table. Are X and
Y independent? Explain.

c) Find E(XY ).

MATH 464: TEST 2 MAKE UP 5

(4) Let X and Y be independent, discrete random variables. Suppose
that

fX(k) = fY (k) = p(1−p)k for all k = 0,1,2, · · ·
for some 0 < p < 1. Show that for any n ≥ 0,

P(X = k |X + Y = n) =
1

n + 1
for any 0 ≤ k ≤ n.

6 SPRING 2013

(5) Let X and Y be independent random variables. Suppose X is
Poisson with parameter λ > 0 and Y is geometric with parameter
0 < p < 1. Let Z = 2X −3Y .

a) Find MZ(t).

b) Find the variance of Z.

MATH 464: TEST 2 MAKE UP 7

(6) a) Let X be a continuous random variable with pdf

fX(t) = exp[−t−exp(−t)] for all t ∈ R .
Find FX(x).

b) Find the real number a for which

fX(x) =



a(x + 1) −1 ≤ x ≤ 0
a(x−1)2 0 ≤ x ≤ 1

0 otherwise

is the probability density function for a continuous random variable
X.

8 SPRING 2013

(7) Let X be a continuous random variable with uniform distribution
on [1,4]. Consider the new random variable

Y = (X −2)2 + 1 .

a) Graph Y on the range of X.

b) Find the cdf, FY (y), of Y .

c) Find the pdf, fY (y), of Y .