Department of Economics
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 Y1 and Y2 be random variables with the same variance. Show that Y1+Y2 and Y1-Y2 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 >?