Like binomial and Bernoulli random variables.
A Poisson random variable is another type of discrete random variable, because again with Poisson,
we're focusing on a countable number of outcomes.
So we'll always treat Poisson variables as discrete, not continuous.
Now the difference between a Poisson random variable and a binomial random variable.
If we remember about the binomial random variable, we indicate a probability of success and then we're
looking for the probability of K successes in n trials.
So a binomial random variable is always indicated this way B of NP or sometimes capital X of NP, where
N is the number of trials for performing.
That's always a fixed number of trials and P is the probability of success on any particular trial.
And when we calculate probability, we're looking at the probability of a certain number of successes
in those end trials.
So if we're running an equals 20 trials and our probability of success is 30% or 0.3, then we could
go on to calculate probability around seeing any number of successes from K equals zero successes all
the way up to K equals 20 successes.
Since we're running an equals 20 trials.
But with the Poisson variable, we have this idea that the rate of occurrence of some event or object
or something that's going on per unit of something else we call lambda, this Greek letter lambda,
and then we're looking for or we're investigating probability of a certain number of occurrences or
successes over one unit where these two units are referring to the same thing here.
So an example could be the number of planes that fly over our house per hour of the day where the occurrence
there is a plane flying over the house and the unit is time in particular ours.
And so if we say that Lambda is equal to ten planes per hour, then we could investigate the probability
of any number of planes flying over the house in one hour.
But we can pick any kind of occurrence.
And this unit doesn't have to be a unit of time.
For instance, we could count the number of dogs that we pass per city block that we walk, and maybe
on average we pass 1.5 dogs for every block that we walk.
And then we could investigate the probability of passing any number of dogs, zero dogs, three dogs,
17 dogs in any one city BLOCK.
So again, this occurrence can be really anything we can imagine.
And these units here can be distance, they can be area volume time.
Any kind of occurrence per unit could in theory, be modeled by a Poisson random variable as long as
we meet a certain set of conditions.
And those conditions are these five.
So again, we're looking for occurrences in an interval.
We have to be counting a certain number of occurrences or events over some other interval unit measurement.
That's what we described here.
That's the first condition we have to meet.
We have to be able to say that the mean is constant for each interval or each unit that we choose.
So in our earlier example where we talked about planes passing over the House, this second condition
might not hold because maybe planes only fly between certain hours of the day, maybe 6:00 am and 10:00
PM.
And so if we're looking at a 24 hour period, this mean is not constant across each interval.
We only want to define a Poisson random variable when all of the intervals were investigating, as far
as we know, at least theoretically can have the same mean.
We also need the count of occurrences or events in each interval to be independent.
So we need to be able to say that the number of events that we count in one interval in one unit doesn't
affect the count we get in the other intervals.
Our counts need to be independent across intervals, and if the count we get in one interval means that
we're going to get a smaller or larger count in the next interval, then we don't have independent counts.
We wouldn't meet this third condition.
We need to have non-overlapping intervals.
So if we're counting planes passing over the house per hour, we can't overlap our hours.
So if we count planes from 9 to 10 a.m., then the next interval would be 10 to 11 and then 11 to 12.
We can't count planes from 9 to 10 and then treat 930 to 1030 as another interval.
Those intervals would overlap.
And then finally, we have to be able to say that probability is proportional to interval size, which
just means that if we determine that the rate at which planes pass over the House is, let's say, ten
planes per hour, then we need to be able to.
To expect that in any two hour interval we should see 20 planes pass over the House.
In a three hour interval, we should see 30 planes pass over the House.
So that probability that mean or rate lambda should scale up or scale down as we scale up or scale down
the interval size.
If all of these things hold, then we can model the situation with a Poisson random variable and its
associated Poisson distribution, and we can use those two things to calculate probability of a certain
number of K successes per unit or per interval.
Now, when it comes to calculating probability, here's what that looks like.
Our probability formula for the probability of K successes in one unit or one interval is lambda to
the power of K, where K is an exponent here multiplied by e to the negative lambda.
So negative lambda is the exponent.
This value e here is Euler's number.
Technically it's a constant.
It's equal to about 2.7 approximately, but it's an infinite decimal number.
Any calculator you use will have a button for Euler's number and then we're dividing by K factorial.
So just to see what this looks like, let's say we're taking an example where we have a customer support
team that receives a certain number of email support tickets per hour and they're responsible for processing
those support tickets.
Maybe we'll say that they receive on average four support tickets per hour such that Lambda is equal
to four, and we then want to investigate probability around the team receiving zero support tickets
in an hour, one support ticket in an hour to support tickets, etc..
Well, if we first check to see that this is a Poisson process, let's bring back our criteria here.
We are, in fact, looking for occurrences in an interval.
We're looking for a certain number of email support tickets per hour.
So we have email support tickets in an interval of time.
That's the kind of situation we're modeling.
So that checks out.
We're saying based on data that we have, that the mean is constant in each interval.
Maybe we have a global company and on average we do get about four support tickets per hour and that
doesn't fluctuate significantly from one hour to the next.
So we meet that condition.
We'll say that the count in each interval is independent because the support tickets we receive in one
hour don't necessarily indicate that we'll receive more or less support tickets in the next hour.
So the interval counts are independent, Our intervals are non overlapping because we're going to look
at each hour of the day without overlapping those hours.
So we'll start maybe at the 8:00 hour in the morning.
Then we'll look at the 9:00 hour, the 10:00 hour, 11:00 hour.
We're not going to overlap our intervals.
So we'll hold to that and we'll say that probability is proportional to interval size, because if we're
getting about four support tickets per hour, we can expect eight support tickets every 2 hours, 12
support tickets every 3 hours, roughly.
So we'll say that we meet this condition as well.
So that's kind of the idea there.
If we say that this situation can be modeled by a Poisson random variable, then here's what probability
looks like for this situation.
So starting to calculate probability, what we want to look at is the probability that this customer
support team gets zero support emails in an hour.
Well, in that case, K is equal to zero.
We already said that on average the support team receives Lambda equals four support tickets per hour.
So with lambda equal to four and K equal to zero, we make substitutions into our probability formula
and we say that we get four to the zero multiplied by E to the negative four divided by zero factorial.
Remember that zero factorial is equal to one.
We can use a calculator to find this value.
If we round to four decimal places, it's approximately equal to 0.0183 or just under 2%.
So there's a slightly less than 2% chance that the customer service team will receive exactly zero support
emails when on average they receive four in an hour.
So slightly less than 2% chance that they'll receive zero in any given hour.
And then we keep everything else the same.
We just change the value of K from 0 to 1 and then to two and then to three.
And these are the probabilities that we get that are associated with the service team receiving exactly
one customer support ticket in any given hour.
Exactly.
Two tickets.
Exactly three tickets.
And what you can see is that as we get closer to the lambda equals four mean value, the probability
is rising.
So since lambda was equal to four, the further we get away from four, the smaller probability we expect
to see.
So the probability that they get zero tickets is just.
Under 2%.
The probability that they get one ticket is a little over 7%.
The probability they get two tickets is about 14 and one half percent.
And the probability that they get three tickets is about 19 and a half percent.
And then if we calculate probability for exactly four customer service emails and then five, six,
seven, eight, we see that the probability that they get four emails is about nine and a half percent.
And then just like we saw on this side of lambda equals four, on the other side of lambda equals four,
we see probability start to drop off as well.
And if we plot these probabilities in a distribution, what we see is the Poisson distribution or the
probability mass function that's associated with this discrete random variable, this Poisson discrete
random variable.
And it makes sense that if Lambda is equal to four, we see these higher probability values clustered
around that mean value of lambda equals four.
And then on both sides of lambda equals four, the probability tapers off or decreases in both directions.
Now realize here that on the left hand side of lambda equals four.
We stopped at zero because of course it doesn't make sense to calculate the probability that the customer
service team receives negative one emails in an hour.
That's nonsensical.
The fewest emails that they can receive is zero emails.
So that's sort of the left hand bound of our probability distribution.
But on the right hand side, the distribution is unbounded.
So we could have actually continued to calculate the probability that they get nine emails, ten emails,
11 emails, 12 emails, etc. on to infinity, and we could have found smaller and smaller and smaller
probability values forever.
But because those values get so insignificant at a certain point, it makes sense just to cut it off.
So this last value that we calculated here was the probability that K is greater than eight.
We calculated the probability that they receive exactly eight emails, and then instead of calculating
the probability that they receive nine emails, remember that because probability always sums to one,
we took one and then we subtracted all of these values that we calculated up through K equals eight.
So we took one -0.0183 -0.0733 -0.1465, etc. Continuing to subtract values all the way until we got
to this last value and we subtracted 0.0298.
And the result of subtracting all of these values from K equals zero to K equals eight from one was
0.0 to 1 three.
So the remaining probability, the probability that they receive nine or more emails in any given hour,
the probability that they receive nine emails or ten emails or 11 emails all the way on to literally
an infinitely large number of emails, all that probability summed up together is a little more than
0.02 or 2%.
So there's just over a 2% chance that they receive nine or more emails.
And we've summarized that here with this last line, and we've also summarized it here in the distribution
to show the probability that they receive nine or more emails in any given hour.
So those are the conditions under which we would use a Poisson random variable and how it differs from
the binomial random variable we looked at earlier.
And then this is the formula we use to calculate probability of K successes in any of the intervals
that we defined, and then how to take all of those probability calculations and use them to create
the Poisson distribution for the Poisson random variable.