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Version A
Room B123, 5:00pm  6:00pm, November 19, 2004
M´arton Bal´azs
NAME:
1.
Assume a reservoir of 10 000 gallons gets Flled every time a rainfall happens, and has
10 000 e

t/
2
gallons of water
t
time after the last rainfall (where
t
is measured in weeks). Times
passing between consecutive rainfalls are independent, and the time
T
between two rainfalls is
an exponentially distributed random variable with parameter
1
2
(weeks

1
).
(a)
(30 points) Let
X
= 10 000 e

T/
2
be the amount of water in the reservoir just when the
next rainfall begins. Then
X
is a random variable. Compute its distribution function and
density function. What kind of random variable is it? What is its expectation?
(b)
(30 points) What is the probability that there are no rainfalls in June (we can assume that
June is precisely 4 weeks long)? What is the expected number of rainfalls in June? What
is the probability that there are at least three rainfalls in June?
1
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 Fall '05
 BALAZS
 Math, Probability

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