The Poisson Distribution

The Poisson distribution is simply the count of random events in a fixed window of time or space, like the number of calls a help desk gets in an hour or typos on a page. It's named after Siméon Denis Poisson, the French mathematician who worked it out. The formula looks odd, but it has only three pieces: $\lambda^k$ (grows with how many events you're asking about), $e^{-\lambda}$ (the chance of zero events, a fixed starting point) and $k!$ (which divides out the orders the $k$ events could come in). You'll mostly just plug in $\lambda$ and $k$, so bear with the notation.

When it applies: events are independent, happen one at a time, and occur at a constant average rate. Calls to a help desk, typos per page, meteors per hour.

Where it comes from. Here's the connection to the binomial you already know. Chop an hour into a huge number $n$ of tiny slices, so small that each holds at most one call, with chance $p = \frac\lambda n$. Now the number of calls is binomial with $n$ trials. Let $n$ grow and $p$ shrink with $np = \lambda$ held fixed, and the binomial formula turns into $\frac{\lambda^ke^{-\lambda}}{k!}$. That's also why, with many trials and a small $p$, the binomial is close to Poisson with $\lambda = np$.

Scaling the interval: if calls average 3 per hour, they average 6 per 2 hours. Change $\lambda$ to match the interval in the question.

Mean equals variance. For a Poisson count both are $\lambda$, so $\sigma = \sqrt\lambda$. It makes sense from the binomial picture: the variance $np(1 - p)$ is almost exactly $np = \lambda$ when $p$ is tiny.

A common mistake. It's tempting to use $\lambda = 3$ for a two-hour question because "the rate is 3." The rate is 3 per hour. $\lambda$ is always the expected count for the interval you're asking about, so for two hours it's 6.

Call centers staff by Poisson arrivals, web servers are sized for Poisson request rates, and epidemiologists use it for counts of rare diseases.