Poisson Counts: Interval Rates, Probabilities, and Moments

All flashcards

What is a Poisson experiment?

An experiment that counts occurrences in a fixed interval when events happen independently at a known average rate.

What does $X$ represent in the Poisson model?

$X$ is the number of occurrences in the interval of interest, so it can take values $0,1,2,3,\ldots$.

How do you adjust a Poisson mean to a shorter interval?

Multiply the known mean by the ratio of the new interval length to the original interval length.

What does $X\sim P(\mu)$ mean?

$X$ is a random variable with a Poisson distribution, and $\mu$ is its mean for the stated interval. The parameter may also be written as $\lambda$.

When can the Poisson distribution approximate a binomial distribution?

When the number of independent trials is large and the success probability per trial is small; the Poisson mean is $\mu=np$.

Quiz questions

1. If $X\sim P(\mu)$, which expression gives the probability of exactly $x$ occurrences?Answer: $\frac{e^{-\mu}\mu^x}{x!}$
2. For a Poisson random variable $X$, how can $P(X>k)$ be calculated?Answer: $1-P(X\leq k)$
3. What is the variance of a Poisson random variable with mean $\mu$?Answer: $\mu$
4. What is the standard deviation of a Poisson random variable with mean $\mu$?Answer: $\sqrt{\mu}$
5. A Poisson mean of 8 occurrences describes repeated equal-length intervals. What does that mean imply?Answer: Eight is the average count, not a guaranteed count in each interval.