typed answer using computer is required.
strict due date/time.
MATH 464
HOMEWORK 7
SPRING 20
1
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
1
2 SPRING 2013
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.