3.6 The Poisson Probability Distribution

Contents

• 3.6.1 The Poisson Distribution as a Limit
• 3.6.2 The Mean and Variance of X
• 3.6.3 The Poisson Process
• EXERCISES Section 3.6 (79-93)

Introduction

• Distributions:

  • Binomial distribution
  • Hypergeometric distribution
  • Negative binomial distribution
  • Derived from: - Experiments consisting of trials or draws
    • Application of laws of probability to various outcomes

• Poisson distribution:

  • No simple experiment as a basis
  • Obtained through:
    • Certain limiting operations (to be described shortly)

Poisson distribution

A discrete random variable $X$ is said to have a Poisson distribution with parameter $\mu \left(\mu >0\right)$ if the pmf of $X$ is $p\left(x;\mu \right)=\frac{e^{-\mu }\cdot {\mu }^x}{x!}x=0,1,2,3,\dots$

Poisson distribution:

• Symbol $\mu$:
  • Represents the Poisson parameter
  • In fact, the expected value of $X$
• Letter $e$ in the pmf:
  • Represents the base of the natural logarithm
  • Numerical value: approximately 2.71828
  • Comparison with other distributions:
    • Unlike binomial and hypergeometric distributions:
    • Spreads probability over all non-negative integers
    • Infinite number of possibilities
• Legitimacy of the Poisson pmf $p(x;\mu )$:
  • Not immediately obvious that it specifies a legitimate pmf
• Conditions:
  • $p(x;\mu )>0$ for every possible $x$ value
  • Requires $\mu >0$
• Normalization:
  • $\sum p(x;\mu )=1$
  • Consequence of the Maclaurin series expansion of $e^{\mu }$:
  • $$\begin{array}{rl}{e}^{\mu }& =1+\mu +\frac{{\mu }^2}{2!}+\frac{{\mu }^3}{3!}+\cdots \\ & =\sum _{x=0}^{\infty }\frac{{\mu }^x}{x!}\end{array}\text{\hspace{1em}(3.18)}$$
• Manipulation of (3.18):
  • Multiply both extreme terms by $e^{-\mu }$
  • Move quantity inside summation:
  • Result:
    1 = \sum_{x = 0}^{\infty} \frac{e^{-\mu} \cdot \mu^{x}}{x!}
• Additional resources:
  • Appendix Table A.2 contains Poisson cdf $F(x;\mu )$ for:
  • $\mu =0.1,0.2,\dots ,1,2,\dots ,10,15,20$
  • Many software packages provide:
    • $F(x;\mu )$ and $p(x;\mu )$ upon request
• EX 3.38 traps