Understanding the Poisson distribution - illustrated using Excel.
Hi. I'm Dr Nic, and in this video I'm going to tell you about
the Poisson Probability Distribution.
This video builds on other videos, with links in the description below.
In particular it will help if you understand what a random
variable is, particularly discrete random variables,
and what a probability distribution is.
The Poisson distribution is named after Simeon Poisson
who first devised it. In the US it is pronounced PoissON. In UK English we say Poisson.
It is the French word for fish. So the inventor was Mr Fish.
Monsieur Poisson sounds much more exotic.
A Poisson distribution is used when you want
to model how likely it is for a number of discrete or individual events to happen in
a given period of time.
A common application of the Poisson distribution is
in waiting line or queueing models.
The Poisson distribution is a good approximation
for the number of people arriving in a given time period,
phone calls at a call-centre or jobs arriving at a print queue.
The individual events are discrete, they cannot happen at the same time and are independent.
A Poisson distribution is defined by a single parameter, lambda.
Lambda is called the rate and is stated as the number of events per time period.
Lambda is also the mean of the distribution and the variance.
The standard deviation is the square root of lambda.
The Poisson distribution can also be used
for the number of events in other specified intervals such as distance or area,
such as number of flaws in a length of fabric or potholes in a road
or the number of lumps of cookie dough in a scoop of ice cream.
Here is an example applying the Poisson distribution.
Helen's cat. Helen's cat Gareth likes to bring Helen dead mice.
On average Gareth presents Helen
with about one mouse per week.
How likely is it that Gareth will bestow Helen with 4 mice in one week?
This scenario is well suited to being modeled
by a Poisson distribution. It is about the number of occurrences of an
event - in this case number of mice brought in
In a fixed unit of time - a week.
We will model this as a Poisson distribution
with a rate, lambda, of one mouse per week.
We want to know the probability of four mice in one week.
I will use Excel in this instance to show you the probabilities.
Here is the Excel formula for finding the
probability of exactly four mice.
The function is Poisson.Dist which has three inputs.
X is the number of events, in this case 4 mice.
The mean is 1 , that is one mouse per week.
Then we need to state true or false as to whether this is cumulative.
If we wanted the probability of all possibilities up to and including 4 mice we would say true.
But as we want the probability of exactly 4 mice, we say false.
The result tells us that the probability of exactly four mice in one week is 0.015.
Possible but not likely.
Let's look at the whole probability distribution
for Poisson with a lambda of 1.
The probability of receiving no mice in one
week is 0.368 and the probability of one mouse is 0.368.
There are decreasing probabilities for more than one mouse.
We say that the distribution is skewed to the right.
Theoretically using this model it is possible
to get more than 6 mice in a week, but the probability is extremely small.
Now we will look at the shape of the Poisson probability distribution
for various values of lambda.
Here is the table and graph for lamda = 1
This is one mouse per week that we started with.
Say Gareth only brings in one mouse every four weeks.
Then our rate will be 1 divided by 4 or 0.25 per week.
In this case you can see that the probability
of no mice in a week is much greater.
And with a lambda of 0.1, the probability
of no mice in one week is greater still. This all makes sense.
When we look at increasing values of lambda, we see something interesting.
As lambda increases the graph gets more and more symmetric about the mean, and starts
to look like a normal distribution.
Here is another example of calculating Poisson probabilities.
Luke sells ice creams in his ice cream stand.
From his sales records, Luke knows that on sunny days he serves about 30 customers in an hour.
Luke has found that a Poisson distribution
is good for modelling customer arrivals. He models the customer arrivals as a Poisson
distribution with a rate, lambda, of 30 per hour.
Luke wants to know how likely it is that fewer than three customers will arrive in the next
five minutes. He wants to find the Probability that X < 3.
As the time period he is interested in is five minutes, he needs to convert to number
of customers in five minutes rather than number of customers per hour.
This is a step that people sometimes miss when calculating Poisson probabilities.
30 customers per hour converts to 2.5 arrivals in a 5 minute period.
Lambda is 2.5. Using Excel, the function is =POISSON.DIST(2,2.5,TRUE).
This will give Luke the probability of less than or equal to 2 arrivals.
Less than or equal to 2 is the same as less than 3 when we are talking about discrete
random variables. We do not wish to include 3 in the interval.
The probability is 0.544.
There is a 54% chance that there will be fewer
than 3 customers in the next five minutes.
On the graph we can see that Luke is most
likely to have 1,2,or 3 customers in any five minute period.
The probability of no customers in any five minute period is 0.08 or 8%.
For more examples of applying the Poisson and binomial distributions see the links in
the description below. Requirements and assumptions of the Poisson distribution.
The individual events cannot happen at the same time.
Individual events are independent.
The probability that an event occurs does
not depend on how long it has been since the last event.
For example, whether Gareth has just brought in a mouse, or it has been two weeks since
his last little present, the probability that he will bring one in is the same. This is
possibly an instance where the model differs from the reality.
Answering questions using the Poisson distribution.
There are three main things you need to be aware of when using the Poisson distribution.
1. You need to make sure that the lambda or
rate or mean applies to the period of time you are interested in
2. You need to be careful of whether you are looking at the probability of an individual
event or a range of events.
3. You need to think about whether your probability is meant to include the given value.
This video was a quick overview of the Poisson distribution.
Check in the description below for other related videos.
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