The Two point distribution

(a) Unsymmetrical case. $X$ has a two-point distribution if it takes two values $x_1,x_2$ with

with cdf $F(x)=0$ for $x<x_1$, $F(x)=p$ for $x_1\le x<x_2$, and $F(x)=1$ for $x\ge x_2$.

Mean: $E(X)=x_1p+x_2(1-p)=px_1+qx_2$, writing $q=1-p$. Variance: starting from $V(X)=E(X^2)-(E(X))^2=(x_1^2p+x_2^2q)-(px_1+qx_2)^2$, expanding and simplifying (using $p+q=1$) collapses to the compact form

$V(X)=pq(x_2-x_1)^2.$ …