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FINAL EXAM, Math 471
(1)
[11 pts] Give an example of a sequence of random variables
{
X
n
}
that converges in probability,
but not with probability one. Be explicit about what the event space
S
is and how
X
i
maps
S
to the
real numbers. Also, be sure to prove that your example indeed converges in probability and doesn’t
converge with probability one.
(2)
[9 pts] One has 100 lightbulbs whose lifetimes are independent exponential random variables
with mean 5 hours. If the bulbs are used one at a time, with a failed bulb replaced immediately with
a new one, what is the approximate probability that there is still a working bulb after 525 hours?
(3)
[10 pts] Suppose
{
A
n
}
is a sequence of events in the event space
S
. Let
B
be the set of all
outcomes
ω
that belong to inﬁnitely many of the
A
i
. The Borel-Cantelli Lemma says that
P
(
B
) = 0
if
∑
i
P
(
A
i
)
<
∞
. Use the Borel-Cantelli Lemma to prove that a sequence of random variables
{
X
n
}
converges to
Y
with probability one if

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