Poisson Process 1

Introduction to Poisson Processes and the Poisson Distribution.

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

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

point on the street at
any given point in time?

And you want to figure out
the probabilities that a

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

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

variable that essentially
represents what you care about.

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

time, let's say, in an hour.

And your goal is to figure out
the probability distribution of

this random variable and then
once you know the probability

distribution then you can
figure out what's the

probability that 100 cars pass
in an hour or the probability

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

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

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

we're going to study the
Poisson distribution.

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

we have to make:

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

than any other hour.

And we know that that's
probably false.

During rush hour in a real
situation you probably

would have more cars than
at another rush hour.

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

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

actually, no.

I shouldn't do a day.

We have to assume that every
hour is completely just like

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

really no differentiation from
one second to the other in

terms of the probabilities
that a car arrives.

That's a little bit of a
simplifying assumption that

might not truly apply to
traffic, but I think we

can make that assumption.

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

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

cars will pass in the next.

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

affect or correlate or somehow
influence the number of cars

that pass in the next.

That they're really
independent.

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

we have to model out some
type of a distribution.

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

distribution is maybe we
can estimate the mean.

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

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

to be a pretty good estimator
for the actual mean

of our population.

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

of this random variable.

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

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

the letter lambda.

You know, this could
be 9 cars per hour.

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

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

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

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

that's a pretty good estimate.

So that's what you have there.

And let's see what we could do.

We know the binomial
distribution.

The binomial distribution tells
us that the expected value of a

random variable is equal to the
number of trials that that

random variable's kind
of composed of, right?

Before, in the previous videos
we were counting the number

of heads in a coin toss.

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

probability of success
over each toss.

This is what we did with
the binomial distribution.

So maybe we can model
our traffic situation

something similar.

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

So maybe we could say lambda
cars per hour is equal

to-- I don't know.

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

whether a car passes
in a given minute.

So there are 60 minutes
per hour, so there

would be 60 trials.

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

those trials, if we modeled
this as a binomial distribution

would be lambda over
60 cars per minute.

And this would be
a probability.

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

that this is a binomial
distribution.

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

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

distribution, so the
probability that our random

variable equals some
given value, k.

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

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

So n would be 60.

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

probability of success.

So the probability that a
car passes in any minute.

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

successes we need.

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

that no cars pass,
to the n minus k.

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

There are 60 minus k minutes
where no car passed.

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

you have 60 intervals and you
say this is a binomial

distribution.

And you'd probably get
reasonable results.

But there's a core issue here.

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

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

Or more than one car
passes in a minute?

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

car passes in a minute.

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

if 5 cars pass in that minute.

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

I just have to get
more granular.

Instead of dividing it
into minutes why don't I

divide it into seconds?

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

intervals I'll do
3,600 intervals.

So the probability of k
successful seconds, so a second

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

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

passes in any given second.

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

number seconds in an hour.

We're going to
have k successes.

And these are the failures,
the probability of a failure

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

And this would be even a
better approximation.

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

situation where 2 cars
can come within a half a

second of each other.

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

We just have to get more
and more granular.

We have to just make
this number larger and

larger and larger.

And your intuition is correct.

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

Poisson distribution.

And this is really interesting
because a lot of times people

give you the formula for the
Poisson distribution and you

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

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

distribution and the binomial
distribution really did come

from kind of the common
sense of flipping coins.

That's where everything
is coming from.

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

as-- let me change colors.

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

right here, the number of
intervals approaches infinity

that this becomes the
Poisson distribution.

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

tools in our belt.

So the first is something that
you're probably reasonably

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

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

equal to e to the
ax-- no sorry.

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

let's make a little
substitution here.

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

n is equal to a over x.

And then what would be
x would equal to na.

x times 1 is equal
to n times a.

And so the limit as x
approaches infinity,

what does a approach?

a is-- sorry.

As x approaches infinity
what does n approach?

Well n is x divided by a.

So n would also
approach infinity.

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

substitution the limit as n
approaches infinity of 1

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

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

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

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

of that to the a.

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

of this and then take
that to the a power.

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

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

that to the a.

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

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

This is how we got to e.

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

and larger n's here
and you'll get e.

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

power, so it's equal
to e to the a.

So hopefully you pretty
satisfied that this limit

is equal to e to the a.

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

probably actually do the
proof in the next video.

The other tool kit is to
recognize that x factorial over

x minus k factorial is equal to
x times x minus 1 times x

minus 2, all the way down
to times x minus k plus 1.

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

abstract we've ever written it.

I can give you a couple of--
and just so you know, they'll

be exactly k terms here.

1, 2, 3-- So first term, second
term, third term, all the

way, and this the kth term.

And this is important to
our derivation of the

Poisson distribution.

But just to make this in real
numbers, if I had 7 factorial

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

times 5 times 4 times
3 times 3 times 1.

Over 2 times-- no sorry.

7 minus 2, this is 5.

So it's over 5 times 4
times 3 times 2 times 1.

These cancel out and you
just have 7 times 6.

And so it's 7 and then
the last term is 7 minus

2 plus 1, which is 6.

In this example, k was 2 and
you had exactly 2 terms.

So once we know those two
things we're now ready

to derive the Poisson
distribution and I'll do

that in the next video.

See you soon.