IMPORTANT: For Quiz, questions a, b, and c are on the exam. Also for quiz, instead of leaving answers in exponential form, convert it to decimal.

Section 5.5 Poisson Distribution and Poisson Process

A discrete random variable X is said to have the Poisson distribution w/ parameter $\mu > 0$ if its probability mass function is given by: $P(X = x) = \frac{e^{-\mu}\mu^x}{x!}$ for x in all positive integers

The mean and the variance of X are both given by $\mu$, ex: $E(X) = \mu = Var(X)$

IT IS IMPORTNT TO USE THE PROBABILITY NOTATION ( P(X = x) ). STANDARD FOR 411 CLASS.

Proof for Var(X) = mu

$E(X) = 0P(X=0) + 1P(X=1) + 2P(X=2) + … $ … $= e^{-\mu}{\mu}e^{\mu} = {\mu}e^0 = \mu$

Must know proof for the final exam. Proof located on page 285.

Poisson Experiments

Located on page 286.

Definition: They are experiments involving such randon variables.

X must be the number of outcomes occuring during a given time interval (or some plan region with area t, or a given space region with volume t.)

There is no upperbound, since we don’t know the upper bounds for this experiment (could be anything).

The experiment is derived from the Poisson Process.

What is a Poisson Random Variable?

The number of outcomes occurring during a Poisson experiment. Its discrete probability distribution is called the Poisson distribution. Number of outcomes is: $E(X) = \mu = {\lambda}t$

t is specified time, distance, area, or volume of interest. $\lambda$ = rate of occurrence of outcomes (ex: per unit time, length, area, region).

There are three properties for the Poisson process: 1. 2. 3.

Example of poisson random variable in real life applications: cars entering the parking lot at 8am in the morning. ALso look at Prussian Horses example.

IMPORTANT TO KNOW POISSON RANDOM VARIABLE, AS THEY ARE IN THE QUIZ AND FINAL EXAM.

Ex: $P(X <= 2) = P(X=0) + P(X=1) + P(X=2)$

$P(X = x) = P(x;{\lambda}t) = dpois(x, {\lambda}t)$

  • Says that the probability that exactly x outcomes occur in a time interval of length t.

$P(X <= x) = P(x;{\lambda}t) = ppois(x, {\lambda}t)$

Must know what d-binom and p-binom for exam.

Poisson Approximation to the binomial distribution

Let X be a binomial random variable w/ parameters n (=number of trials) and p (=probability of success in each trial). When n approaches infinity and p approaches 0:

For poisson approaximation, we must specify X to the problem (look at page 295).

look at correlation coefficient.