The Poisson Distribution Discrete Random Variables 1 Some

The Poisson Distribution Discrete Random Variables 1

Some Special Discrete Distributions � Sometimes we can use specific distributions to model situations � Some ◦ ◦ ◦ important discrete distributions are: Binomial Poisson Geometric Negative Binomial Hypergeometric 2

Binomial →Poisson � As the number of trials (n) in a binomial experiment increases to infinity while the binomial mean (np) remains constant, the PMF of the binomial distribution becomes the PMF of the Poisson distribution. 3

The Poisson Distribution � In general, a Poisson random variable X is the number of events (counts) per interval. � Often data this is based on observation or historical � Examples: ◦ ◦ Power outages per year. Weeds per Acre of farm land Fleas per Dog Cardiovascular deaths per year in a given county

Poisson Distribution � 5

Poisson Cautions � It is important to use consistent units in the calculation of Poisson means, variances, and probabilities � We must consider the rate we are given and the interval we are asked about. � Ex. Suppose I am given that I receive 4 emails per hour. ◦ How many emails do you expect in the next 2 hours? �This means we expect µ = 8 emails in 2 hrs. ◦ How many emails do you expect in the next half hour? �This means 2 emails per half hour ◦ How many emails do you expect in the next minute? � 4/60 = 1/15 emails per minute 6

Binomial VS. Poisson and beyond � Poisson ◦ The number of events (x) likely to happen on a fixed interval with rate λ � Binomial ◦ Probability of x successes in a fixed number of trials � What if our numbers get really big? ◦ We can approximate these distributions with the Normal ◦ We will focus on this with the binomial, but it can also be done in a similar manner with the Poisson

Poisson Example � 8

Poisson Example cont… �

Another Poisson Example �