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Unformatted text preview: day before or was ﬁnished being repaired 68 CHAPTER 2. DISCRETETYPE RANDOM VARIABLES the day before or that it is the ﬁrst day, the machine is up on the given day with probability 0.999
and it goes down with probability 0.001. If a machine goes down on a given day, then the machine
stays down for repair for a total of ﬁve days. The two machines go up and down independently.
We would like to estimate the mean time until there is a day such that both machines are down.
This is a challenging problem. Can you show that the mean time until outage is between 200
and 400 years? Here is a reasonably accurate solution. Given both machines were working the day
before, or were just repaired, or it is the ﬁrst day, the probability that at least one of them fails
on a given day is about 0.002. That is, the waiting time until at least one machine goes down is
approximately geometrically distributed with parameter p = 0.002. The mean of such a distribution
is 1/p, so the mean time until at least one of the machine...
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 Spring '08
 Zahrn
 The Land

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