Poisson Distribution

Parameters

• $\lambda$: rate.

A random variable is said to be Poisson distributed if the probability of getting exactly $k\in \mathbb{N}$ occurrences is described by the following

Probability Mass Function

$$\begin{array}{rrcl}{p}_X(\cdot ,n,p):& \mathbb{N}& \to & \mathbb{R}\\ & k& \to & \frac{{\lambda }^k{e}^{-\lambda }}{k!}.\end{array}$$

Moments

• Expected value: $E[X]=\lambda$,
• Variance: $\text{Var}(X)=\lambda$.

The Poisson distribution is an appropriate model if the following assumptions are true:

• $x$ is the number of times an event occurs in an interval and $x$ can take values in $\mathbb{N}$.
• The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently.
• The average rate at which events occur is independent of any occurrences. For simplicity, this is usually assumed to be constant, but may in practice vary with time.
• Two events cannot occur at exactly the same instant; instead, at each very small sub-interval, either exactly one event occurs, or no event occurs.

Relationship with other Distributions

Let $X$ be a discrete random variable following a Binomial distribution with parameter $n$. Then, $X$ converges to the Poisson distribution with parameter $\lambda =np$, when $n\to \infty$ and $p\to 0$.