Interview question: Let X a standard normal random variable. What can you say about the correlation between the positive part and the negative part of X i.e $corr(X^+, X^-)=corr(max(X,0), min(X,0))$?
I tried to calculate with explicit formula based on normal pdf but $Var(X^+)$ seems to be difficult to compute and does not seem to be what the interviewer expects.
Thanks!
I cannot see a simple way which differ from direct calculations. First, usually negative part is not $\min(X,0)$, but $X^{-}=-\min(X,0)$. And this can only change a sign of correlation. The expectations and variances can be calculated directly: $$ \mathbb EX^{+}=\mathbb EX^{-}=\int_0^\infty t\frac{1}{\sqrt{2\pi}}e^{-t^2/2}\,dt = \frac{1}{\sqrt{2\pi}}. $$ $$ \mathbb E X^{+}\cdot X^{-}=0 $$ since $X^{+}\cdot X^{-}=0$. $$ Var(X^{+})=Var(X^{-})=\mathbb E\left[(X^{+})^2\right]-\left(\mathbb EX^{+}\right)^2 = \int_0^\infty t^2\frac{1}{\sqrt{2\pi}}e^{-t^2/2}\,dt - \frac{1}{2\pi} = \frac12-\frac{1}{2\pi}. $$ So $$ corr(X^+, X^-)=\dfrac{0-\frac{1}{2\pi}}{\frac12-\frac{1}{2\pi}} = \frac{1}{1-\pi}. $$