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Summary: Homework 1
Due September 1, Friday
1. a. Let X1 and X2 be independent and identically distributed exponential
random variables with parameter , show that X1 + X2 has the following
density function
f(t) = 2
e-t
t
b. If X and Y are independent exponential random variables with respec-
tive means 1/µ1 and 1/µ2, compute the distribution of Z = min(X, Y ).
What is the conditional distribution of Z given that Z = X?
2. Show that for a nonnegative random variable X with distribution func-
tion F, E[X] =
0
¯F(x)dx.
3. Show that for any sequence of events E1, E2, E3, · · ·, P(
i=1
Ei)
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