probability

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MATH 464

HOMEWORK 6

SPRING 2013

The following assignment is to be turned in on
Thursday, March 7, 2013.

1. Suppose that in a certain state the license plates have three letters fol-
lowed by 3 numbers. If no letter or number can be repeated, how many
license plates are possible?

2. A club has 50 members. The club need to form two committees, one with
8 members and one with 7 members. How many ways can this be done if
no one is allowed to serve on two committees at the same time?

3. 6 students, 3 boys and 3 girls, line up in random order for a photograph.
What is the probability that the boys and girls alternate?

4. A fair coin is tossed 10 times. What is the probability of 5 heads? What
is the probability of at least 5 heads?

5. I have a television with 50 channels. On a certain evening, 12 are showing
sit-coms, 17 are showing reality shows, 15 are showing movies, and the
remaining 6 are showing something else. If I randomly pick 5 of the channels
and look at what is showing, what is the probability that I see:

a) exactly 2 movies, 1 sit-com, and 2 reality shows?

b) at least one movie?

c) only sit-coms and reality shows?

6. Consider a usual deck of cards. Draw five cards at random. What is the
probability you get:

a) ”four of a kind” or four cards of the same rank?

b) a ”full-house” or three cards of the same rank and two cards of the same
rank?

c) ”three of a kind” or three cards of the same rank, but you do not have a
”full-house”?

7. I have 4 friends and 15 cookies. How many ways are there to:
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a) give away all the cookies with no constraints?

b) give away all the cookies making sure every friend gets at least 2 cookies?

c) give away some (or none) of the cookies with no constraints?

8. A round table has n seats. n people are seated at random around the
table. Fred, who is sitting at the table, dislikes two of the people. Let X be
the number of neighbors of Fred whom he dislikes. Find the p.m.f. of X.
(Note that X can only be 0, 1, 2. )

MATH 464

HOMEWORK 7

SPRING 20

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3

The following assignment is to be turned in on
Thursday, March 28, 2013.

1. Consider the following experiment: Roll 2 fair, four sided dice. Consider
the following discrete random variables:

X = the number of odd dice.

Y = the number of even dice.

Z = the number of dice showing a 3 or a 4.

Clearly, each of X, Y , and Z have range {0, 1, 2}.
a) Find fX,Y (x,y). Give your answer in tabular form.

b) Determine whether or not X and Y are independent.

c) Find E(XY ).

d) Repeat exercises a) – c) above for the random variables Y and Z.

2. Suppose that X and Y are discrete random variables and that you know
the joint probability mass function of X and Y is:

fX,Y (x,y) = α
x+y+1 for x,y = 0, 1, 2 with some α > 0.

Find E(XY ) and E(Y ).

3. Let X and Y be independent discrete random variables. Suppose we
know that

E(X) = −2, E(X2) = 5, E(X3) = 10, and E(X4) = 50

and

E(Y ) = −1, E(Y 2) = 5, E(Y 3) = −13, and E(Y 4) = 73

a) Let Z = 2X + Y . Find the mean and variance of Z.

b) Let W = Y 2 − 2Y X2. Find the mean and variance of W .

4. Let X and Y be independent discrete random variables. Suppose X is a
Poisson random variable with parameter λ > 0 and Y is a Poisson random
variable with parameter µ > 0. Show that the random variable Z = X + Y
is also a Poisson random variable and determine its parameter. Hint: You

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may want to use the formula:

(1 + x)n =

n∑
k=0

(
n

k

)
xk for any integer n ≥ 1 and real number x.

5. Suppose you have an unfair coin with probability p for heads. Consider
the following 2 stage experiment: First, flip the coin until you get a heads.
Then, flip the coin again until you get a tails. Let X be the discrete random
variable counting the total number of flips in this 2 stage experiment.

a) Find the mean and variance of X. Hint: It may be useful to write X as
the sum of 2 random variables. If you do, label and describe carefully each
of these random variables.

b) Let Y be the number of heads minus the number of tails in this 2 stage
experiment. Find the mean and variance of Y .

6. Let X and Y be independent discrete random variables. Suppose that
each of them is geometric and that you know E(X) = 2 and E(Y ) = 3.

a) Find the joint probability mass function of X and Y .

b) Find the probability that X + Y ≤ 4.

c) Consider W = min{X,Y} and Z = max{X,Y}. Find the joint probabil-
ity mass function of W and Z.

7. Let X1, X2, · · · , X100 be independent discrete random variables. Suppose
that each of them is a Poisson random variable with λ = 2. Consider

X =
1

100

100∑
j=1

Xj

which is sometimes called the sample mean. Find the mean and variance of
X.

8. Suppose you have an unfair coin with probability p for heads. Do an
experiment where you flip this coin N times, and let N be a random number
which is Poisson with parameter λ > 0. Assume that N is independent of
the outcomes of the flips. Let X be the number of heads. Let Y be the
number of tails. Find the probability mass functions for X and Y and use
your result to show that X and Y are independent.

MATH 464

HOMEWORK 8

SPRING 2013

The following assignment is to be turned in on
Thursday, April 4, 2013.

1. Let X be a Poisson random variable with parameter λ > 0.

a) Find the moment generating function for X.

b) Use your result above to find the mean of the random variable Z =
2X3 − 3X2 + X.
c) Consider n ≥ 1, independent, discrete random variables X1, X2, · · · ,
Xn, and suppose that each are Poisson with parameter λ > 0. Let Z =
X1 + X2 + · · · + Xn. Find the pmf of Z.

2. Let X be a negative binomial random variable with parameters n and p.
Calculate the variance of X.

3. Let X be an exponential random variable with parameter λ > 0.

a) Let t ≥ 0 and calculate P(X ≥ t).
b) Let s,t ≥ 0 and calculate P(X ≥ s + t|X ≥ s). (You can compare your
answer to this question with your answer to problem #5 on homework #5.)

4. The gamma function is defined by

Γ(w) =

∫ ∞
0

xw−1e−x dx

for all w > 0. In terms of this function, a continuous random variable X
(with parameters w > 0 and λ > 0) is defined by setting

fX(x) =

{
λw

Γ(w)
xw−1e−λx if x > 0,

0 otherwise.

and declaring that X has probability density function fX(x). (fX is called
the gamma distribution with parameters w > 0 and λ > 0.)

a) Show that X is a continuous random variable by showing that∫
R
fX(t) dt = 1

for all values of w > 0 and λ > 0.
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b) Show that for any w > 1,

Γ(w) = (w − 1)Γ(w − 1)
Use your result to calculate Γ(n) for any integer n ≥ 2.
c) Compute the mean and variance of this random variable X.