**Temple University
Department of Economics**

Economics 615

Homework 1

**Probability, Random Variables and Distributions**

**1.** The game of craps is played as follows. A player tosses
two dice until the player either wins or loses. The player wins on the first toss if
the player gets a total of 7 or 11; the player loses if the total is 2, 3 or 12. If
the player gets any other total on the first toss, then that total is called the
point. The player then tosses the dice repeatedly until either a total of 7 or the
point is obtained. The player wins if the player gets the point and loses if the
player gets a 7. What is the player's probability of winning?

**2.** Let Y be a random variable with probability density function
f(y). The mode of of the distribution of Y, assuming it exists and is unique, is
that value of y that maximizes f(y). Find the mode in the following case.

**3.** Let F(x,y) be the distribution function of X and Y. Show that

for all real numbers a>b, c<d.

**4.** Let Y_{1} and Y_{2} be random variables with
the same variance. Show that Y_{1}+Y_{2} and Y_{1}-Y_{2}
are uncorrelated.

**5.** Suppose f(y|x) is defined by

f(y=-1|x=-1) = f(y=3|x=-1) = ½

f(y=0|x=0) = f(y=4|x=0) = ½

f(y=0|x=1) = f(y=2|x=1) = ½

and that f(x=-1) = f(x=0) = f(x=1) = 1/3

5.a. Find E(Y|x=-1), E(Y|x=0), and E(Y|x=1)

5.b. Find E(XY|x=-1), E(XY|x=0), and E(XY|x=1)

5.c. Find E(X), E(Y) and E(XY)

5.d. Is Y independent of X?

5.e. Are X and Y uncorrelated?

6. Consider the random variables X and Y with means and , variances and , and covarinace . Suppose Y is the return on a stock and X is the return on a bond such that >and >. Finally let W be a Bernoulli random variable distributed independently of X and Y, where P(W=1)=p and P(W=0)=1-p, and 0<p<1. Consider the random portfolio Z=WX+(1-W)Y.

6.a. Find .

6.b. Which is larger or ?

6.c. Find .

6.d. How large must **-**be in order for **>**?