Poisson Distribution Explained

• The Poisson distribution deals with the probability of a certain number of events occurring randomly within a specific interval.

Properties of a Poisson Experiment

• The number of outcomes in one interval/region is independent of outcomes in other disjoint intervals/regions.
• The probability of an outcome in a short interval/small region is proportional to the length/size and doesn't depend on outcomes outside it.
• The probability of multiple outcomes in a very small interval approaches zero.

Poisson Random Variable

• The number $X$ of outcomes in a Poisson experiment is a Poisson random variable.

Poisson Distribution Formula

• Given $\lambda$ as the average number of outcomes, the probability of $x$ outcomes is: $P(X = x) = \frac{e^{-\lambda}\lambda^x}{x!}$, where $e = 2.71828...$ and $x = 0, 1, 2,...$

Mean and Variance

• For a Poisson random variable $X$ with parameter $\lambda$:
  • Mean: $\mu = \lambda$
  • Variance: $\sigma^2 = \lambda$

Example 1: Accidents Per Day

• Accidents occur at a rate of approximately 2 per day.
  • Probability of no accidents: $P(X = 0) = \frac{e^{-2}2^0}{0!} = e^{-2} = 0.1353$
  • Probability of two or more accidents: $P(X \ge 2) = 1 - P(X < 2) = 1 - P(X = 0) - P(X = 1) = 1 - \frac{e^{-2}2^0}{0!} - \frac{e^{-2}2^1}{1!} = 1 - e^{-2} - 2e^{-2} = 1 - 3e^{-2} = 0.5940$

Example 2: Bus Arrival Times

• Buses arrive every 20 minutes. A person arrives at 7:22 am
  • Probability of waiting no more than 5 minutes: $P(X \le 5) = \int_{0}^{5} \frac{1}{20}e^{-\frac{x}{20}}dx = 1 - e^{-\frac{5}{20}} = 1 - e^{-0.25} = 0.2212$
  • Probability of waiting at least 12 minutes: $P(X \ge 12) = \int_{12}^{\infty} \frac{1}{20}e^{-\frac{x}{20}}dx = e^{-\frac{12}{20}} = e^{-0.6} = 0.5488$

Exercise 1: Bacteria in Ground Beef

• Mean: 5 bacteria per square in
  • The probability that there are no bacteria is required
  • The probability that there are at least 2 bacteria is required

Exercise 2: Cars at a Tollbooth

• Cars arrive at an average rate of 5 per minute, with interarrival times being exponentially distributed.
  • Probability that the interarrival time between cars is more than 30 seconds
  • Probability that the interarrival time between cars is less than 10 seconds