expectation of product of sums of normally distr. r.v.

Question

Let $Z_1$ and $Z_2$ be i.i.d. standard normally distributed. $X_1=Z_1+Z_2$ and $X_2=Z_1-Z_2$.

Apparantly E[|$X_1|*|X_2|$] = E$[|Z_1|*|Z_2|]$. Why?

Answer 1

This is a special case of the more general phenomenon that two i.i.d. variables with normal distribution, considered as coordinates in the plane, yield a rotationally invariant distribution over the plane. This is due to the fact that the product of the exponentials is the exponential of the sum of the exponents, so the combined exponent is $x_1^2+x_2^2=r^2$, with rotational symmetry. Your transformation is a scaling by $\sqrt2$ and a rotation through $\pi/4$, and the rotation leaves the distribution invariant.

This rotational invariance is also used in calculating the integral over a one-dimensional Gaussian (squaring it yields a two-dimensional Gaussian, and the Jacobian factor $r$ allows the integral to be evaluated), and in efficiently generating independent Gaussian random variables two at at time.