Let x y be independent exponential random variables

5. Let X, Y be independent exponential random variables with mean 1. Find the joint
density of ( X + Y, X - Y ). Write down clearly the ranges for the variables.

6. Let the joint probability density of X, Y is given by
f
X,Y
( x, y ) =
(
Cxy if 0 < x < y < 1
0 otherwise.
Find C and the marginal density of X . Also find P (2 X ≤ Y ).
R
y
R
1
0
1 3
dy =
1
8
. So C = 8.
7. Suppose that the expected number of accidents per week at an industrial plant is 5.
Suppose also that the numbers of workers injured in each accident are independent
random variables with a common mean of 3. If the number of workers injured in each
accident is independent of the number of accidents that occur, compute the expected
number of workers injured in a week.
8. A fair coin is tossed 100 times, and suppose Y denotes the total number of times
heads turned up. Let X denote the number of tails among the first 50 tosses. Find
V ar ( Y | X = 0). ( Hint : Check if X and Y - 50 + X are independent).