Geometric & Poisson Distributions Cheat Sheet

Key Facts

• For a geometric random variable $X$, $X$ is the trial number of the first success, so possible values are $X=1,2,3,\ldots$.
• The geometric probability formula is $P(X=k)=(1-p)^{k-1}p$, where $p$ is the probability of success on each independent trial.
• For a geometric distribution, the mean is $E(X)=\frac{1}{p}$ and the variance is $\operatorname{Var}(X)=\frac{1-p}{p^2}$.
• The geometric cumulative probability for success on or before trial $k$ is $P(X\le k)=1-(1-p)^k$.
• For a Poisson random variable $X$, $X$ counts the number of events in a fixed interval, so possible values are $X=0,1,2,3,\ldots$.
• The Poisson probability formula is $P(X=k)=\frac{e^{-\lambda}\lambda^k}{k!}$, where $\lambda$ is the mean number of events in the interval.
• For a Poisson distribution, the mean and variance are both equal to $\lambda$, so $E(X)=\lambda$ and $\operatorname{Var}(X)=\lambda$.
• If the rate is $r$ events per unit and the interval length is $t$, then the Poisson parameter is $\lambda=rt$.

Vocabulary

Geometric distribution

A probability distribution that gives the chance that the first success occurs on trial $k$ in repeated independent trials.

Poisson distribution

A probability distribution that gives the chance of observing $k$ events in a fixed interval when events occur at an average rate $\lambda$.

Success probability

The value $p$ is the probability that one trial results in a success.

Rate parameter

The value $\lambda$ is the average number of events expected in the chosen interval.

Expected value

The expected value $E(X)$ is the long-run average value of the random variable $X$.

Independence

Independence means the outcome of one trial or event does not change the probability of another.