- [Narrator] The Poisson distribution
is an interesting distribution
and we're gonna talk about that in this screen cast.
Before we get started on this,
I just want you all to say that word,
it's pronounced "pwa-sewn."
It's a French word meaning fish,
so that's how you pronounce it.
♪ Les Poissons, les Poissons ♪
♪ How I love les Poissons ♪
♪ Love to chop and to serve little fish ♪
- [Narrator] The Poisson distribution
is when we have a continuous interval
but we have discreet events occurring
so discreet events occurring on a continuous interval.
This interval can be something like time.
So, I show that here.
Example I'm gonna be going over today
in this screen cast is going to be earthquakes.
Earthquakes do not occur on a consistent regular period.
It's not like we have a certain number
of earthquakes every year
or a probability of an earthquake every day.
Instead what happens is if we start our time
we might have an earthquake, you know, later this year,
we might have another one here,
we might have one immediately after that.
And so, they're discrete events.
The events are earthquakes
but they're occurring in a continuous interval
and that continuous interval is time.
Whenever you have discrete events happening
on continuous interval, you should think
about using the Poisson distribution.
Another example where we might have a continuous
interval would be physical distance.
You know, this might be potholes in a road.
So you're traveling down this road, so there's your car,
and occasionally you might hit a pothole.
So the events, the discrete events are potholes.
So you might hit a pothole there.
It might be a while before you hit another one
but then there might be a bunch in the next little bit.
And so their random occurrences,
discrete events in a continuous interval.
The interval does not have to be time or distance.
It can be area, so maybe this is a silicon wafer.
Occasionally, on that silicon wafer
you're gonna have defects
so you could have a defect there,
there, you might have two over here.
The event, the discrete event is a defect
but it's in a continuous area.
It's a continuous interval.
And so this would be an example of where
you might want to consider the Poisson distribution.
Another example of a continuous interval but would be volume
and let's just say we have a suspension of cells
and we have living cells and dead cells.
And in this volume you might have discrete events
that are cells that are dead.
So you might have dead cells that
are in this continuous interval.
The discrete events are dead cells
and the interval is volume.
So let's just sum this up.
The Poisson distribution is applied
to situations where the rate of occurrence
of discrete events is specified and it's known.
Usually the rate is a number of per unit time
as I talked about, but it can also be in distance,
it can be area and it can actually be volume.
This rate of occurrence is given by this symbol lambda.
So lambda is actually the number of events per interval.
That could be cells per cubic centimeter,
that could be flaws per square centimeter.
That could be potholes per mile,
that could be earthquakes per year.
The random variable that we model using
the Poisson distribution is the number
of events, X, that occur in the continuous interval
and the probability function is given here
and this is known as the Poisson distribution.
It's important to know that in order for you
to model this you have to know the interval.
This is given a capital T
because we oftentimes model intervals over time
but this interval could also, again,
be distance, area, or volume.
But T is your continuous interval.
If you are looking at the number of earthquakes
in two years, your interval would be two years.
You can't estimate things with
the Poisson distribution unless you know
the interval that you're considering.
The main in variance interestingly,
are exactly the same for the Poisson distribution.
Based upon historical data, we know that
in the last 40 years there have been 200 earthquakes.
The first part of this is what is the mean
or the expected value for the number
of earthquakes in the next two years?
Well, to determine this we can determine a lambda.
Lambda is the background rate of occurrence
and this is, we have 200 earthquakes in 40 years,
so lambda is gonna be five earthquakes per year.
Now this is not the mean, the expected value,
or the mean also depends upon T and our time interval.
For this problem, the time interval is two years
and the mean, going back to a previous slide,
is given simply by Lambda times the interval.
So in this case, our expected value
or our mean, is going to be two years
and we expect about five earthquakes per year.
We would expect in two years to find
that there have been 10 earthquakes.
Let's go through the second part.
The second part of the problem is asking
what is the probability that there will be
exactly six earthquakes in the next two years?
Just like some
of the other discrete probability distribution functions,
the X value here is a discrete variable
and the output F of X is the probability
of exactly X number of occurrences.
To answer this, let's first look at our background rate.
We calculated this as five earthquakes
per year or events per year.
In the previous part of the question.
Our time interval here is two years.
So the mean or lambda T that goes
into this function should be 10, 10 events.
And a lot of times we leave off the earthquakes
and that implies that we have events,
10 events occurring in the next two years
and if we plug lambda T into the Poisson distribution,
we get the following
and we can plug this into our calculator.
We get this to be about 0.063, which means,
in the next two years there's a 6.3% probability
that there's gonna be exactly six earthquakes.
Let's move along to part C.
What is the probability that there will be at
least six earthquakes in the next two years?
Just like some of the other discrete distributions
we've looked at we can put together a histogram
or a column chart of this.
Graphically, at least six earthquakes
in the next two years would be the probability
of six, the probability of seven.
And we sum everything to the right of this, right?
Now, keep in mind that in contrast
to the binomial distribution, we can keep going forever.
We could potentially have a hundred
and that's gonna be very small, but,
at least you have to consider summing everything above that.
That's very hard to do because you would have
to add F of six, plus F of seven, plus F of eight,
all the way up to very large numbers.
What's better to do
and way easier is to calculate F of five.
That's accumulative up to and including five.
And then we subtract that from one
because the sum of all of these bars is gonna be one.
And so, that's what you can do.
You can take one minus the sum of F of zero
through F of five, which is also capital F of five.
Let me show you how to do this in Excel
because it's way easier in Excel.
I'll turn it to the V,
you can just plug this into your calculator.
So I'm over here in Excel.
Let's just fill out a couple things.
We had five earthquakes per year,
that's our long term average.
If we're looking at two years and I'm actually going
to add in here a cell for our mean.
Our mean is just the product of lambda
and our interval, that's 10.
I'm just gonna name that mean real quick
and let's go ahead and use
the Poisson dot disfunction in Excel.
The first argument is the number
of events that you're interested in.
So that's gonna be X here.
The mean is our mean that we calculated above.
So I type in our mean that we've named
and I'm gonna use here
the non-cumulative distribution function.
That means the probability of getting exactly
zero earthquakes in the next two years is very, very small.
It's nearly zero because of how I set this up.
I can easily copy that down.
And this is telling us that the the probability
of getting exactly X number of earthquakes per year.
And you see that we get here, 6.3%
which is what we got in part B when we worked it earlier.
Now to answer the question
of what's the probability that you'll get at
least six earthquakes in the next two years?
We would have to sum the individual probabilities
of these six through 15, but I just stopped at 15.
This would keep going and there's, again,
there's a probability that you would have 20 earthquakes
in the next two years, 30 and so on.
So, really you would have to add all the bars
beyond this up until, theoretically, infinity.
Instead, we can use
the cumulative Poisson distribution function
and again, this is gonna be one minus F of five
because we're being asked at least six.
So we're gonna take one minus, up to and including five,
which is the cumulative Poisson distribution up to
and including five.
And so I can just do one minus the Poisson distribution
of five 'cause we're interested in X equals five.
The mean is our mean that we calculated
and instead of false, I'm gonna use true.
So it's accumulative.
And when we do that, it's telling us
that there's a 93.3% chance that we will have six
or more earthquakes in the next two years.
Hopefully this screen cast give you a better idea
of the Poisson distribution
and how to implement it in Microsoft Excel.