ORIE 3500/5500 – Engineering Probability and Statistics II
Fall 2010
Assignment 7
Problem 1
Four runners,
A, B, C
and
D
are organized in two teams,
with runners
A
and
B
forming Team 1, and runners
C
and
D
forming
Team 2. All four runners will run the same distance. Suppose that the
running times of the 4 runners are all exponentially distributed, with
respective means
μ
A
= 10,
μ
B
= 12,
μ
C
= 8 and
μ
D
= 14 minutes.
Assume also that all four running times are independent.
(
a
) What is the distribution of the
actual
fastest time in each team?
(
b
) What is the probability that the winner of the race (the fastest
one of the four runners) will come from Team 2?
Problem 2
Derive the moment generating function of the binomial
distribution and use it to derive the mean and the variance of a binomial
random variable.
Problem 3
If
X
is a Poisson random variable with mean
λ
, show
that
p
X
(
i
) first increases and then decreases as
i
increases, reaching is
maximum when
i
is the largest integer less than or equal to
λ
.
Problem 4
You arrive at a bus stop at 10 0’clock, knowing that
the bus will arrive at some time uniformly distributed between 10 and
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 '08
 WEBER
 Probability, Probability theory, binomial random variable

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