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Let's say you're some type of
traffic engineer and what

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you're trying to figure out is,
how many cars pass by a certain

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point on the street at
any given point in time?

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And you want to figure out
the probabilities that a

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hundred cars pass or 5
cars pass in a given hour.

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So a good place to start is
just to define a random

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variable that essentially
represents what you care about.

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So let's say the number of cars
that pass in some amount of

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time, let's say, in an hour.

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And your goal is to figure out
the probability distribution of

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this random variable and then
once you know the probability

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distribution then you can
figure out what's the

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probability that 100 cars pass
in an hour or the probability

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that no cars pass in an hour
and you'd be unstoppable.

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And just a little aside, just
to move forward with this

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video, there's two assumptions
we need to make because

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we're going to study the
Poisson distribution.

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And in order to study it's
there's two assumptions

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we have to make:

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That any hour at this point
on the street is no different

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than any other hour.

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And we know that that's
probably false.

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During rush hour in a real
situation you probably

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would have more cars than
at another rush hour.

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And you know, if you wanted to
be more realistic maybe we do

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it in the day because in a day
any period of time--

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actually, no.

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I shouldn't do a day.

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We have to assume that every
hour is completely just like

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any other hour and actually,
even within the hour there's

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really no differentiation from
one second to the other in

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terms of the probabilities
that a car arrives.

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That's a little bit of a
simplifying assumption that

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might not truly apply to
traffic, but I think we

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can make that assumption.

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And then the other assumption
we need to make is that if a

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bunch of cars pass in one hour
that doesn't mean that fewer

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cars will pass in the next.

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That in no way does the number
of cars that pass in one period

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affect or correlate or somehow
influence the number of cars

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that pass in the next.

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That they're really
independent.

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Given that, we can then at
least try using the skills

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we have to model out some
type of a distribution.

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The first thing you do and I'd
recommend doing this for any

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distribution is maybe we
can estimate the mean.

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Let's sit out on that curb and
measure what this variable is

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over a bunch of hours and then
average it up, and that's going

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to be a pretty good estimator
for the actual mean

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of our population.

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Or, since it's a random
variable, the expected value

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of this random variable.

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Let's say you do that and you
get your best estimate of the

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expected value of this random
variable is-- I'll use

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the letter lambda.

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You know, this could
be 9 cars per hour.

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You sat out there-- it could
be 9.3 cars per hour.

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You sat out there over hundreds
of hours and you just counted

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the number of cars each hour
and you averaged them all up.

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You said, on average, there are
9.3 cars per hour and you feel

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that's a pretty good estimate.

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So that's what you have there.

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And let's see what we could do.

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We know the binomial
distribution.

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The binomial distribution tells
us that the expected value of a

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random variable is equal to the
number of trials that that

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random variable's kind
of composed of, right?

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Before, in the previous videos
we were counting the number

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of heads in a coin toss.

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So this would be the number
of coin tosses, times the

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probability of success
over each toss.

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This is what we did with
the binomial distribution.

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So maybe we can model
our traffic situation

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something similar.

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This is the number of cars
that pass in an hour.

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So maybe we could say lambda
cars per hour is equal

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to-- I don't know.

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Let's make each experiment or
each toss of the coin equal to

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whether a car passes
in a given minute.

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So there are 60 minutes
per hour, so there

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would be 60 trials.

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And then, the probability that
we have success in each of

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those trials, if we modeled
this as a binomial distribution

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would be lambda over
60 cars per minute.

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And this would be
a probability.

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This would be n, and this would
be the probability, if we said

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that this is a binomial
distribution.

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And this probably wouldn't be
that bad of an approximation.

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If you actually then said,
oh, this is a binomial

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distribution, so the
probability that our random

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variable equals some
given value, k.

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You know, the probability that
3 cars, exactly 3 cars pass in

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an given hour, we would
then be equal to n.

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So n would be 60.

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Choose k, and you know,
I have 3 cars times the

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probability of success.

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So the probability that a
car passes in any minute.

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So it'd be lambda over
60 to the number of

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successes we need.

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So to the kth power, times the
probability of no success or

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that no cars pass,
to the n minus k.

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If we have k successes we have
to have 60 minus k failures.

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There are 60 minus k minutes
where no car passed.

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This actually wouldn't be that
bad of an approximation where

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you have 60 intervals and you
say this is a binomial

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distribution.

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And you'd probably get
reasonable results.

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But there's a core issue here.

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In this model where we model it
as a binomial distribution,

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what happens if more than
one car passes in an hour?

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Or more than one car
passes in a minute?

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The way we have it right now
we call it a success if one

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car passes in a minute.

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And if you're kind of counting
it counts as one success, even

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if 5 cars pass in that minute.

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So you say, oh, OK Sal, I
know the solution there.

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I just have to get
more granular.

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Instead of dividing it
into minutes why don't I

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divide it into seconds?

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So the probability that I have
k successes-- instead of 60

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intervals I'll do
3,600 intervals.

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So the probability of k
successful seconds, so a second

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where a car is passing at that
moment out of 3,600 seconds.

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So that's 3,600 choose k, times
the probability that a car

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passes in any given second.

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That's the expected number of
cars in an hour divided by

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number seconds in an hour.

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We're going to
have k successes.

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And these are the failures,
the probability of a failure

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and you're going to have
3,600 minus k failures.

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And this would be even a
better approximation.

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This actually would not be so
bad, but still, you have this

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situation where 2 cars
can come within a half a

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second of each other.

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And you say, oh, OK Sal,
I see the pattern here.

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We just have to get more
and more granular.

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We have to just make
this number larger and

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larger and larger.

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And your intuition is correct.

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And if you do that you'll
end up getting the

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Poisson distribution.

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And this is really interesting
because a lot of times people

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give you the formula for the
Poisson distribution and you

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can kind of just plug in
the numbers and use it.

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But it's neat to know that it
really is just the binomial

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distribution and the binomial
distribution really did come

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from kind of the common
sense of flipping coins.

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That's where everything
is coming from.

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But before we kind of prove
that if we take the limit

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as-- let me change colors.

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Before we proved that as we
take the limit as this number

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right here, the number of
intervals approaches infinity

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that this becomes the
Poisson distribution.

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I'm going to make sure we have
a couple of mathematical

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tools in our belt.

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So the first is something that
you're probably reasonably

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familiar with by now, but I
just want to make sure that the

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limit as x approaches infinity
of 1 plus a/x to the x power is

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equal to e to the
ax-- no sorry.

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Is equal to e to the a and now
just to prove this to you,

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let's make a little
substitution here.

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Let's say that n is equal
to-- let me say 1 over

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n is equal to a over x.

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And then what would be
x would equal to na.

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x times 1 is equal
to n times a.

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And so the limit as x
approaches infinity,

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what does a approach?

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a is-- sorry.

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As x approaches infinity
what does n approach?

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Well n is x divided by a.

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So n would also
approach infinity.

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So this thing would be the same
thing as just making our

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substitution the limit as n
approaches infinity of 1

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plus-- a/x, I made the
substitution as 1/n.

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And x is, by this
substitution, n times a.

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And this is going to be the
same thing as the limit as n

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approaches infinity of 1 plus
1/n to the n, all

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of that to the a.

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And since there's no n out here
we could just take the limit

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of this and then take
that to the a power.

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So that's going to be equal to
the limit as n approaches

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infinity of 1 plus 1/n to the
nth power, all of

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that to the a.

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And this is our definition, or
one of the ways to get to e if

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you'd watch the videos on
compound interest and all that.

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This is how we got to e.

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And if you tried it out on your
calculator, just try larger

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00:09:03,460 --> 00:09:07,260
and larger n's here
and you'll get e.

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This inner part is equal to e,
and we raised it to the a

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00:09:12,010 --> 00:09:14,060
power, so it's equal
to e to the a.

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So hopefully you pretty
satisfied that this limit

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00:09:16,240 --> 00:09:17,860
is equal to e to the a.

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And then one other tool kit I
want in our belt, and I'll

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00:09:19,860 --> 00:09:22,340
probably actually do the
proof in the next video.

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The other tool kit is to
recognize that x factorial over

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00:09:32,950 --> 00:09:42,860
x minus k factorial is equal to
x times x minus 1 times x

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00:09:42,860 --> 00:09:50,030
minus 2, all the way down
to times x minus k plus 1.

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And we've done this a lot of
times, but this is the most

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00:09:51,880 --> 00:09:53,060
abstract we've ever written it.

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00:09:53,060 --> 00:09:55,580
I can give you a couple of--
and just so you know, they'll

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be exactly k terms here.

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00:09:57,330 --> 00:10:01,700
1, 2, 3-- So first term, second
term, third term, all the

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00:10:01,700 --> 00:10:04,310
way, and this the kth term.

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00:10:04,310 --> 00:10:07,210
And this is important to
our derivation of the

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00:10:07,210 --> 00:10:09,160
Poisson distribution.

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00:10:09,160 --> 00:10:13,870
But just to make this in real
numbers, if I had 7 factorial

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over 7 minus 2 factorial,
that's equal to 7 times 6

208
00:10:20,110 --> 00:10:24,070
times 5 times 4 times
3 times 3 times 1.

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00:10:24,070 --> 00:10:27,360
Over 2 times-- no sorry.

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00:10:27,360 --> 00:10:28,940
7 minus 2, this is 5.

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00:10:28,940 --> 00:10:33,500
So it's over 5 times 4
times 3 times 2 times 1.

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00:10:33,500 --> 00:10:37,190
These cancel out and you
just have 7 times 6.

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00:10:37,190 --> 00:10:40,990
And so it's 7 and then
the last term is 7 minus

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00:10:40,990 --> 00:10:43,045
2 plus 1, which is 6.

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00:10:47,560 --> 00:10:51,290
In this example, k was 2 and
you had exactly 2 terms.

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00:10:51,290 --> 00:10:53,230
So once we know those two
things we're now ready

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00:10:53,230 --> 00:10:55,710
to derive the Poisson
distribution and I'll do

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00:10:55,710 --> 00:10:58,415
that in the next video.

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See you soon.