Introduction

The Poisson distribution is named after Simeon Denis Poisson, a French mathematician, engineer, and physicist who made many scientific advances. It is used to model the number of occurrences of an event per space or time, for example

  • Number of weeds per square foot
  • Number of calls per hour
  • Number of defects per square inch

Probability Distribution

The probability distribution for \(X\), a Poisson random variable with mean \(\lambda\) is \[p(x) = P(X=x) = \frac{e^{-\lambda}\lambda^x}{x!} \text{ for } x = 0,1,\cdots\] where \(e \approx 2.71828\)

In R, \(p(x) = \text{ dpois}(x,\lambda)\)

The cumulative distribution function for \(X\), a Poisson random variable with mean \(\lambda\) is \[F(x)=P(X \le x) = \sum_{k=0}^{x} \frac{e^{-\lambda}\lambda^k}{k!}\] In R, \(F(x) = \text{ ppois}(x,\lambda)\)

Example: Customers arrive at a busy checkout counter at an average rate of three per minute. Find the probability that in any given minute there will be

  • exactly one arrival
  • at most one arrival
  • at least two arrivals

\(X\) = number of customers per minute with \(\lambda=3\)

  • \(P(X=1) = \frac{e^{-3}3^1}{1!} = \text{ dpois}(1,3) = 0.1494\)
  • \(P(X \le 1) = \frac{e^{-3}3^0}{0!} + \frac{e^{-3}3^1}{1!} = \text{ ppois}(1,3) = 0.1991\)
  • \(P(X \ge 2) = 1 - P(X \le 1) = 1 - 0.1991 = 0.8009\)

Summary

If \(X\) has a Poisson distribution with mean \(\lambda\) then

\[p(x) = P(X=x) = \frac{e^{-\lambda}\lambda^x}{x!} = \text{ dpois}(x,\lambda)\] \[F(x)=P(X \le x) = \sum_{k=0}^{x} \frac{e^{-\lambda}\lambda^k}{k!}=\text{ ppois}(x,\lambda)\]