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In this segment, we go
through a quick review

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of a few properties of
the Bernoulli process

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that we already know.

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We start by thinking about the
number of successes or arrivals

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in the first n time slots.

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This is the following quantity.

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At each time we add a
0 or a 1, depending on

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whether we've had
a success or not,

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then by adding those numbers,
we get the total number

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of successes.

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Now we already know that the
number of successes in n trials

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obeys a binomial distribution,
so the probability

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of having k successes is given
by the binomial probabilities.

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And this is a formula that
holds for k equal to 0 up to n,

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which are the possible numbers
for the random variable S.

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For this random variable,
we know the expected value.

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It's n times p.

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And we also know its
variance, which is n times

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p times 1 minus p.

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Another random
variable of interest

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is the time until the
first success or arrival.

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So this is defined
to be the smallest

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i for which the random
variable Xi is equal to 1.

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We have done this
calculation in the past.

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The probability that the first
success appears at time k is

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the same as the probability
that the first k minus 1 trials

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resulted in 0's.

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And then, the k-th
trial resulted in a 1.

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And so the probability
of this is 1 minus p,

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the probability of 0, and
we have k minus 1 of them,

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times p, the probability
that the next trial gives us

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a success.

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And this formula is valid
for k being 1, 2, and so on,

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which is the range
of possible values

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of this random variable T1.

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This is the familiar
geometric distribution

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that we have dealt with
on several occasions.

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And in particular, we know the
expected value and the variance

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of the geometric
random variable.