> [!NOTE] Lemma > Let $n\in \mathbb{N}$ and $p\in[0,1].$ The function $ 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}$is a [[Probability Mass Function|probability mass function]]. **Proof**: We have $1=(p+(1-p))^n=\sum_{k=0}^n\binom{n}{k}\cdot p^k\cdot(1-p)^{n-k}.$ # Applications A discrete real-valued random variable is said to have [[Binomial Distribution|binomial distribution]] if its probability mass function is given by the above function.