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We will now go through a very
simple example, in which we

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just get to use the formulas
that we have available.

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The Poisson process is a pretty
good model of email arrivals

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at least during a
limited part of the day.

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For example, during
daytime, emails

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may be arriving as a Poisson
crosses with a certain rate.

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But then during
night time, they may

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be arriving as a Poisson
process with a different rate.

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Nevertheless, the
assumption that we will make

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is that, at least
for this problem,

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that emails arrive
as a Poisson process

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with a fixed rate of
five messages per hour.

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What is the mean
and the variance

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of the number of emails
received during a day?

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Well, we have formulas for
the mean and the variance.

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And in this problem, we
have a lambda equal to 5,

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and tau consists of 24 hours.

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So the answer is 5 times 24.

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And this answer applies to both
of the mean and the variance,

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because for the Poisson random
variable, these are the same.

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What is the
probability that we get

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one new message
in the next hour?

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This has to do with the PMF
of the number of arrivals

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during the next
hour, and that PMF

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is given by the
Poisson probabilities.

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We're asking for the
probability of one new message,

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so that k is equal to
1, in the next hour,

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so that tau is equal to 1.

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So we're looking for
this expression here.

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And using also that
lambda is equal to 5,

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we enter those numbers
into this formula.

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And what we obtained is
5 times e to the minus 5.

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Finally, what is the
probability that during each

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of the next three hours,
you'll obtain two messages.

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So this is an event
which is actually

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the intersection
of three events,

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the event of two
messages in this hour,

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two messages in this hour,
and two messages in that hour.

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For the Poisson process, we
have assumed that different time

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intervals are independent
of each other.

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So what we need to
do is to multiply

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the probability of two
messages in this hour

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with the probability of
two messages in that hour,

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and the probability of two
messages in that power.

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On the other hand, for
each one of the hours,

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the probability's
going to be the same,

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so it's enough to take the
probability of two messages

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during this hour, which is in
our notation this quantity,

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and multiply it with
itself three times,

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so we get the third
power of this.

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Now this expression is
equal to the following.

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Lambda times tau is 5.

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k is equal to 2, so
we get 5 squared.

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Then we have an e
to the minus 5 term.

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And k is equal to 2,
so we're dividing by 2.

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And we take the
third power of this.