> [!NOTE] Definition 1 > Let $X$ be a [[Discrete random variables|discrete real-valued random variable]]. Let $\lambda \geq 0.$ Let $n\in \mathbb{N}.$ Then $X$ has a *Poisson distribution* with parameter $\lambda \geq 0,$ denoted $X \sim \text{Poisson}(n,p),$ if its [[Probability Mass Function of Discrete Real-Valued Random Variable|probability mass function]] is given by $p_{X}(x) = \begin{cases}\frac{e^{-\lambda}\lambda^{x}}{ x!}, & x\in \mathbb{N}, \\0, & \text{otherwise.} \end{cases}$where $e^{x}$ denotes the [[Real Exponential Function|exponential function]]. **Note**: By [[Poisson Distribution Probability Mass Function is Probability Mass Function]], $p_{X}$ is indeed a probability mass function. # Properties By [[Binomial Distribution Approximated by Poisson Distribution]], if $X\sim \text{Binom}(n,p)$ then $p_{X}\to p_{Y},$ where $Y\sim \text{Poisson}(np)$ as $n\to \infty.$ By [[Expectation of Poisson Distribution]], if $X\sim \text{Poisson}(\lambda)$ then $\mathbb{E}[X]=\lambda.$ By [[Variance of Poisson Distribution]], if $X\sim \text{Poisson}(\lambda)$ then $\text{Var}(X)=\lambda.$