11. Poisson Distribution

The ________ distribution is used to model discrete data

Poisson

The _______ distribution predicts the pattern in which random events of extremely low probability occur over the course of a very large number of trials

Poisson

_________ _________ _________ can take on any positive integer value, whereas the binomial distribution always has a finite upper limit

→ Examples:

Number of pieces of mail per day

Number of colony forming units in a bacteria culture

Number of industrial accidents in a week

Poisson random variables

The Poisson distribution is a ______ probability distribution, and applies when:

1) The event is something that can be ________ in whole numbers

2) occurrences are __________

3) the average frequency of occurrence for the time period in question is ______

4) it is possible to count how many events have occurred, but meaningless to ask how many events have not occurred

discrete

counted

independent

known

If we let the random variable 𝑋 be the number of events in a given interval, then the mean (expected) number of events per interval is __

ʎ

The probability of observing 𝑘 events in an interval is given by:

We say that 𝑋 follows a Poisson distribution with parameter __

As it turns out, the mean and variance of a Poisson random variable are always ____

→ __ is also the variance of the distribution

ʎ,

equal

ʎ,

If the mean, 𝜆, is a natural number (𝜆 ∈ ℕ+), then there will be ___ modes, 𝜆 and 𝜆 − 1.

two

As 𝜆 increases, the center of the curve moves to the _____

right

As 𝜆 increases (𝜆 ≈ 7), the probability of zero occurrences becomes ________

unlikely

Poisson is (symmetric or asymmetric)?

asymmetric

If the event is rare (p<0.1), then Poisson will approximate the _________ __________

binomial distribution