07 Poisson Distribution

The Poisson distribution used for counting the number of occurrences of an event within a given time period, area, or volume, where the events happen at a constant average rate $\lambda$ (the rate parameter), and the events are independent.

Notation:

We write:

$$X\sim \text{Poission}(\lambda )$$

where:

• $X$ is the number of events (like the number of cars passing a toll booth, or the number of phone calls in an hour).
• $\lambda$ is the average rate of occurrence (e.g., 5 cars per minute).
• $X$ can take any non-negative integer value : $X\in \{0,1,2,3,\dots \}$

Probability Mass Function (PMF) :

The probability of observing exactly $k$ events is given by the Poisson formula:

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

where:

• $k$ : number of events you want the probability for,
• $\lambda$ : average rate of events (mean of the distribution),
• $e$ : Euler’s number ($e\approx 2.71828$)

Key Properties:

• Mean : $\mathbb{E}[X]=\lambda$
• Variance : $\text{Var}(X)=\lambda$

Real life example:

Let’s say the average number of cars passing a toll booth in one hour is $\lambda$=5$\lambda$ = 5$\lambda$=5. The number of cars passing in any given hour follows a Poisson distribution. If we want to know the probability of exactly 3 cars passing the toll booth in the next hour, we use the Poisson formula with $k$=3$k$ = 3$k$=3 and $\lambda$=5$\lambda$ = 5$\lambda$=5.