Named After Siméon-Denis Poisson

 Binomial and Geometric distributions only work when we have Bernoulli trials.  There are three conditions for those.  They happen often enough, to be sure, but a good many situations do not fit those models.  The Poisson Distribution works in a slightly different situation, but different enough that the situations we can describe just shoots through the roof.

 The Poisson distribution is used when we have a discrete random variable and all we know is the following two things:  The average count in a certain timeframe.  That the amount of time it takes the event to happen again is independent.  I think some examples will really clear this up.

 If you know the average number of car wrecks per week/month/year for a specific intersection (or stretch of highway), then it is Poisson.  If you know the average number of defective units produced on an assembly line per hour, then it is Poisson.  If you know the average number of children born in Pocatello, ID each day, then it is Poisson.

 Wikipedia cites some examples in the real world.  The number of soldiers killed by horse-kicks each year in each corps in the Prussian cavalry.  The number of phone calls at a call centre per minute.  Under an assumption of homogeneity, the number of times a web server is accessed per minute.  The number of mutations in a given stretch of DNA after a certain amount of radiation.  The proportion of cells that will be infected at a given multiplicity of infection.

 The math for this involves e (as in the base of the natural log) and involves the factorial (those really excited numbers we talked about yesterday).  The actual formula will only be discussed on Special Topics day, thanks to our calculator.  poissonpdf will be used to find the probability of a specific number of occurrences.  poissoncdf will be used to find the probability that anything from 0 to a specific number of occurrences has happened.

 A Poisson distribution is only defined by the mean. In other words, once we know the mean, we have the Poisson distribution.  The Greek letter we use for this is lambda.  It kind of looks like an upside down lowercase y.  Lambda -> λ  Again, this is the mean of the Poisson distribution, instead of our old standby μ.

 The variance of the Poisson distribution is actually also λ.  Which means that the standard deviation of a Poisson distribution is the square root of λ.  What this really means is that when you add two Poisson distributions together, you can just add the λ values to find the new one.

 What this means is that if a particular intersection averages 2.3 wrecks a month, then it averages 27.6 wrecks a year, and we can just use either figure.  If we know the average per week, we just divide by 7, and booyah…daily average.  So if we know what timeframe the average is for, we can multiply and divide it as necessary to get the one we want.  This only works with Poisson.

 Chapter 16: 15, 25, 32  Chapter 17: 13, 15, 17, 25  Presentations are on Feb 17 th unless you make other arrangements.