Named After Siméon-Denis Poisson

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Presentation transcript:

Named After Siméon-Denis Poisson Poisson Distribution Named After Siméon-Denis Poisson

What’s The Big Deal? 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.

Poisson Distribution 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.

Poisson Distribution Examples 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.

Poisson Distribution Examples 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.

Poisson Distribution 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.

Poisson Distribution 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.

Poisson Distribution 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.

Poisson Distribution 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.