Correlation between min and max of two uniform variables Let $X$ and $Y$ be two i.i.d uniform random variables drawn from $(0,1)$. Let $A$ be $\min(X,Y)$ and $B$ be $\max(X,Y)$, what’s the correlation between $A$ and $B $ ?

$$\overline A=\int_0^1\int_0^1\min(x,y)\,dx\,dy=\int_0^1\left[\int_0^y x\,dx+\int_y^1y\,dx\right]dy=\int_0^1\left[\frac{y^2}2+y(1-y)\right]dy\\
=\frac13.$$
$$\overline B=\int_0^1\int_0^1\max(x,y)\,dx\,dy=\int_0^1\left[\int_0^y y\,dx+\int_y^1x\,dx\right]dy=\int_0^1\left[y^2+\frac{1-y^2}2\right]dy\\
=\frac23.$$
$$\overline{AB}=\int_0^1\int_0^1\min(x,y)\max(x,y)\,dx\,dy=\int_0^1\left[\int_0^y xy\,dx+\int_y^1yx\,dx\right]dy\\
=\int_0^1\int_0^1xy\,dx\,dy=\frac14.$$
Remains to compute the standard deviations.
