Problem: Alternative definitions of Var(X)

Problem

If \(X\) is an arbitrary random variable, discrete or continuous, with expected value \(μ\) then we define

\[Var\left( X \right)=E\left[ {\left( X-μ \right)}^2 \right]\]

Note that if \(X\) is a continuous random variable with range \(\left[ a,b \right]\) and probability density function \(f(x)\) then we have defined

\[Var\left( X \right)=\int_{a}^{b}{ {\left( x-μ \right)}^2f\left( x \right)dx }\]

However, there is no contradiction here since

\[E\left[ {\left( X-μ \right)}^2 \right]=\int_{a}^{b}{ {\left( x-μ \right)}^2f\left( x \right)dx }\]

If \(X\) is a continuous random variable.

Starting from \(Var\left( X \right)=E\left[ {\left( X-μ \right)}^2 \right]\) , show that

\[Var\left( X \right)=E\left( X^2 \right)-μ^2\]

Hint: Expand the parenthesis and use expectation rules.