# Binomial Distribution

We use the Binomial Distribution to answer the following question:

- Given $n$ independent trials and a probability of success $p$ per trial, what is the probability that we have $k$ total successes, i.e. $P(X=k)$?

Suppose $X∼Bin(n,p)$

- Support of $X:{0,1,2,…,n}$
- PMF: $P(X=k)=(kn )p_{k}(1−p)_{n−k}$ where

- $k$ is the number of successes
- $n$ is the number of trials
- $p$ is the probability of success for 1 trial

Expectation and Variance

- $E(X)=np$
- $Var(X)=np(1−p)$