Binomial Distribution

Definition

$X\sim B(n,p)$ where $n$ is the length of bernoulli process, and $p\in [0,1]$ is a probability of success

The number of successes in length $n$ bernoulli process with success probability $p$

Properties

PDF

$f(x)=\left(\genfrac{}{}{0ex}{}{n}{x}\right)p^x(1-p{)}^{n-x},x\in \{0,1,\dots ,n\}$

Mean

$E(X)=np$

Variance

$\mathrm{Var}(X)=np(1-p)$

MGF

$M_X(t)=[(1-p)+p{e}^t{]}^n$

Facts

Let $X_i\sim B(n_i,p)$ be independent random variables following binomial distribution, then $Y=\sum _{i=1}^n{X}_i\sim B(\sum _{i=1}^n{n}_i,p)$