The Poisson Distribution

The probability that there are x occurences of an event (where x is a non-negative integer is given by)

λ is often a rate per unit time, distance, or area. This is a discrete distribution based on counting events. Note as you tend towards large λ (i.e. long times) this distribution becomes like a normal distribution.

Example:

if the events occur on average every 4 minutes, and you are interested in the number of events occurring in a 10 minute interval, you would use as model a Poisson distribution with &lambda= 2.5. (e.g. 10/4 total time window divided by average arrival rate)

In a Poisson distribution:

• the expected value is equal to λ
• standard deviation is square root of λ
• Median is approximately λ - 1/6
• mode is approximately λ - 1/2.

Ignoring the multi day calculus proof of this:

This distribution arises as a consequence of the summation of many rare events.

Examples of environmentally related events governed by this distribution:

• The number of cars that pass through a certain point on a road during a given period of time.

• The number of spelling mistakes a secretary makes while typing a single page.

• The number of phone calls at a call center per minute Poisson statistics govern the behavior of queues! (like the god damn subway line at the EMU)

• The number of roadkill found per unit length of road

• The number of pine trees per unit area of mixed forest.

• The number of soldiers killed by horse-kicks each year in each corps in the Prussian cavalry.

• The number of Cat 5 Hurricanes that pass by a given latitude and longtitude

Some important assumptions

• arrivals/events occur one at a time (hence discrete)

• probability of an arrival/event in a given time interval depends only on the length of the time interval

• the number of arrivals/events in non-overlapping times intervals are independent - that is the arrival of the next event is not correlated with the fact that a previous event has occurred. This is important. If this is not the case, then you can not use this statistical approach.

Example:

Lollipops arrive to the lollipop tester on average every 10 minutes.

The arrival process is Poissonian.

What is the probability of 2 arrivals in the next 15 minutes?

So, x =2 (that's the number of events we are counting)

λ = 1.5 (we expect one per arrival every 10 minutes; total time = 15 minutes; &lambda = 15/10 = 1.5)

P(x) = .251

What about 4 events?

P(x) = .047

The following animation shows the probability distribution of events as a function of λ (shown in upper right hand corner) you can see as λ increases the probability distribution evolves from being skewed to being normal.

Anoter Example:

Determining the probability of a rare event


Assume that the probability of dying from leukemia is described by a Poisson distribution with a mean of 2.3 deaths per 10000 children λ = 2.3

You now have been commissioned to study an apparent increase in leukemia incidences associated with children that live in ToxicWasteVille. Your boss, The Federal Health Authority, has determined that compensation will be based on a 1% criteria - that is, if the probability of the increase in occurrence is less than 1% then they will be compensated.

Your qualitative bias is to be on the side of the children.

You then go to ToxicWasteVille and measure a local death rate of 6 deaths per 10000 children. That is you observed 6 events (X = 6) when the expected event rate was 2.3. Important, you can not observe fractional events!

Will this death rate qualify for compensation?

So, What is the probability of observing a local death rate of 6 deaths per 10000?

P(6) = e^-2.3 2.3⁶ / 6!

P(6) = 0.1003 * 148.04 / 720

P(6) = 0.021 or 2.1% does not meet the compensation criteria

Note: the Poisson distribution would not be used with an infectious disease because the events (people with the disease) would not be independent of each other, i.e. if your partner has an infectious disease the probability of you getting it is higher than some one not in contact with the disease.