# [Math] Expected value of the absolute value of the difference of two random variables

absolute valuenormal distributionprobability

I have to compute the absolute value of an estimator defined as $T_5=\frac{1}{2}E[|X_1-X_2|]$ in order to state if it is unbiased for $\sigma$, where $X$ is distributed as a $N(0,\sigma^2)$.

I am stuck in computing the expected value of the absolute value of the difference of two normal random variables. Any hint?

If $X_1$ and $X_2$ are iid random variables such that $X_1\sim\mathcal N(0,\sigma^2)$ and $X_2\sim\mathcal N(0,\sigma^2)$, then $$X_1-X_2\sim\mathcal N(0,2\sigma^2).$$ If $X\sim\mathcal N(0,\sigma^2)$, then $Y=|X|$ has the half-normal distribution and $$\operatorname EY=\frac{\sqrt2\sigma}{\sqrt\pi}.$$ Hence, we have that $$\frac12\operatorname E|X_1-X_2|=\frac12\frac{\sqrt2\sqrt2\sigma}{\sqrt\pi}=\frac{\sigma}{\sqrt\pi}.$$