Poisson Distribution: Rare Events

What is the Poisson distribution?

The Poisson distribution models how many times a random event occurs in a fixed time interval (or fixed area/volume), when events happen at a constant average rate $\lambda$ and independently of each other.

The formula

$P(X = k) = \frac{e^{-\lambda} \lambda^{k}}{k!}$

where $\lambda$ is the average number of events per interval and $k$ is the actual count you're computing the probability for.

Step-by-step: $\lambda = 5$ calls per hour

What is $P(\text{exactly 3 calls in an hour})$?

Step 1: $k = 3$, $\lambda = 5$.

Step 2: $P(X = 3) = \frac{e^{-5} \cdot 5^3}{3!} = \frac{0.00674 \times 125}{6} = \frac{0.842}{6} = 0.1404$

About a 14% chance of exactly 3 calls.

Key properties

• Mean = $\lambda$ (average rate is the mean — by definition).
• Variance = $\lambda$ too (a unique property of the Poisson).
• Standard deviation = $\sqrt{\lambda}$.
• Shape: right-skewed for small $\lambda$; approximately normal for large $\lambda$.

When to use Poisson

Events that are: (1) random, (2) independent, (3) occurring at a constant average rate. Examples: calls to a call center, typos per page, car accidents per day, meteor strikes per year, website hits per minute.

Poisson vs Binomial

• Binomial: fixed number of trials $n$, each with probability $p$. "Out of $n$ trials, how many succeed?"
• Poisson: events over continuous time/space with rate $\lambda$. "How many events in this interval?"
• Connection: Binomial with large $n$ and small $p$ (rare events) approaches Poisson with $\lambda = np$.

Try it in the visualization

Drag $\lambda$ and watch the bar chart reshape. Small $\lambda$ (rare events) gives a right-skewed distribution. Large $\lambda$ becomes nearly symmetric. Toggle the normal overlay to see the approximation improve.