I'm playing around with the standard deviation of the samples taken from the normal distribution. When calculating the integral
$$\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x) f(y) \sqrt{x^2-\frac{1}{2} (x+y)^2+y^2}dydx$$
Where $$f(x)=\text{PDF}[\text{NormalDistribution}[],x]=\frac{e^{-\frac{x^2}{2}}}{\sqrt{2 \pi }}$$
Mathematica returns 0:
f = PDF[NormalDistribution[]];
Integrate[Sqrt[(x^2 + y^2 - (x + y)^2/2)] f[x] f[y], {x, -Infinity, Infinity}, {y, -Infinity, Infinity}]
(*Outputs 0*)
I know that it can't be 0 because for all $x$, $f(x)>0$, and for all $(x,y)\in\mathbb{R}^2$, $\sqrt{x^2-\frac{1}{2} (x+y)^2+y^2} \geq 0$.
If we replace Integrate with NIntegrate, the output is 0.797884, which seems reasonable, but why does Integrate
break?
PrincipalValue -> True
toIntegrate
. $\endgroup$