Question

What is the formula of variance of a binomial random variable? How it is derived?

Answer 1

Var(X)=np(1-p)
Proof: Let $X=X_1+...+X_n$ where all $X_i$ are independently Bernouli distributed random variables. Since $Var\left(X_i\right)=p(1-p)$, we get
$Var\left(X\right)=Var(X_1+...+X_n)=Var\left(X_1\right)+...+Var\left(X_n\right)=nVar\left(X_1\right)=np(1-p)$
$E\left(X\right)=E(X_1+X_2+...+X_n)=E\left(X_1\right)+E\left(X_2\right)+...+E\left(X_n\right)=np$
Since each $X_1$ Ber(p)
Now there is a result that says that if $Y_1,Y_2,...,Y_K$ arised out of k independent experiments then
$V(Y_1+Y_2+...+Y_K)=V\left(Y_1\right)+V\left(Y_2\right)+...+V\left(Y_K\right)$
We apply that result to get
$V\left(X\right)=V\left(X_1\right)+V\left(X_2\right)+...+V\left(X_n\right)=np(1-p)$