BSAN Poisson Distribution

A discrete random variable that counts the number of events occurring within a fixed interval.

What is a Poisson random variable?

The average number of events or occurrences per unit (the rate)

What does lambda represent in a Poisson distribution?

One parameter, lambda

How many parameters does a Poisson distribution have?

Any nonnegative integer: \(0,1,2,3,\ldots\). There is no fixed upper bound

What values can a Poisson random variable take?

Binomial counts successes in a fixed number of trials using \(n\) and \(p\). Poisson counts events in a fixed interval using the average rate \(\lambda\).

What is the main difference between binomial and Poisson distributions?

The probability of exactly \(x\) events occurring in a fixed interval

What does the Poisson PMF calculate?

The PMF calculates the probability of a specific outcome, while the expected value represents the average number of events.

What is the difference between the PMF and expected value?

Look for wording asking for the probability of exactly a particular number of events in a fixed interval

How do you recognize a Poisson PMF question?

Because Poisson variables have no fixed upper bound, the complement avoids adding infinitely many probabilities

Why is the complement rule useful for Poisson distributions?

The PMF calculates the probability of exactly one value, while the CDF calculates the total probability at or below a value.

What is the difference between a Poisson PMF and CDF?

Lambda doubles, assuming the event rate remains constant.

What happens to lambda when the interval doubles?

Lambda is divided by two, assuming the event rate remains constant

What happens to lambda when the interval is cut in half?

The distribution tends to be positively skewed, with a longer tail extending toward larger values

How does a small lambda typically affect the shape of a Poisson distribution?

The distribution tends to look more symmetric and unimodal

How does a larger lambda affect the shape of a Poisson distribution?

Lambda changes when the interval changes. The value of x represents the specific number of events being asked about

What is the difference between changing lambda and changing x?

Zero events occur across both units

What does "neither of two independent units experiences an event" mean?

The expected number of events increases because \(E(X)=\lambda\)

What happens to the expected number of events when lambda increases?