> [!NOTE] Definition 1 > Let $X$ be a [[Discrete random variables|discrete real-valued random variable]]. Let $n\in \mathbb{N}.$ Let $p\in [0,1].$ Then $X$ has a *binomial distribution* with parameter $n$ and $p,$ denoted $X \sim \text{Binom}(n,p),$ if its [[Probability Mass Function of Discrete Real-Valued Random Variable|probability mass function]] is given by $ p_{X}(x) = \begin{cases} {n \choose x} p^{x} \cdot (1-p)^{n-x}, & x\in \{ 0,1,\dots,n \}, \\ 0, & \text{otherwise} \end{cases}$where ${n \choose k}$ the [[Number of k-Combinations of n Letters|number of k-combinations of n letters]]. **Note**: By [[Binomial 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 Binomial Distribution]], if $X\sim \text{Binom}(n,p)$ then $\mathbb{E}[X]=np.$ By [[Variance of Binomial Distribution]],if $X\sim \text{Binom}(n,p)$ then $\text{Var}(X)=np(1-p).$