 
Summary: Math 501. 4th Homework. Due Wednesday, October 31, 2007.
Homework on "Basic properties of Random variables and Expectation".
1. A real estate agent is selling a house. If the house is sold within the first month after the
home hits the market, the real estate agent makes $3000. If the house is sold within the
second month, the real estate agent makes $2000. If the house is sold within the third
month, the real estate agent breaks even. If the house is not sold by three months, the real
estate agent losses $4000. The probability that the house is sold within the ith month is
1
2i , for i = 1, 2, . . . . What is the real estate agent expected profit?
2. Find two r.v.'s X1 and X2 such that:
(i) For each a < b, P[a < X1 b, a < X2 b] = P[a < X1 b]P[a < X2 b].
(ii) X1 and X2 are not independent r.v.'s.
3. Find two r.v.'s X1 and X2 such that:
(i) Both X1 and X2 have a continuous distribution.
(ii) X1 + X2 does not have a continuous distribution.
4. The random variables X and Y have joint density function
f(x, y) =
3x if 0 x y 2x 2,
0 otherwise.
(i) Find P[Y 1].
