probability

MATH 464

HOMEWORK 7

SPRING 2013

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.