Expected gap between two normals

MediumRandom Variables~6m

Let $X \sim \mathcal N(0,1)$ and $Y \sim \mathcal N(0,4)$ be independent. Compute $\mathbb{E}\,|Y - X|$ . The answer has the form $\left(\dfrac{K}{\pi}\right)^{b}$ for rational $K$ and $b$ ; report $b\cdot K$ .

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$Y-X$ is normal with variance $\operatorname{Var}(Y)+\operatorname{Var}(X)=5$ (variances add).
For $Z\sim\mathcal N(0,\sigma^2)$ , $\mathbb{E}|Z|=\sigma\sqrt{2/\pi}$ . Plug $\sigma=\sqrt5$ .

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Expected gap between two normals

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