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 $ ?
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$$\overline A=\int_0^1\int_0^1\min(x,y)\,dx\,dy=\int_0^1\left[\int_0^y x\,dx+\int_y^1y\,dx\right]dy=\int_0^1\left[\frac{y^2}2+y(1-y)\right]dy\\ =\frac13.$$
$$\overline B=\int_0^1\int_0^1\max(x,y)\,dx\,dy=\int_0^1\left[\int_0^y y\,dx+\int_y^1x\,dx\right]dy=\int_0^1\left[y^2+\frac{1-y^2}2\right]dy\\ =\frac23.$$
$$\overline{AB}=\int_0^1\int_0^1\min(x,y)\max(x,y)\,dx\,dy=\int_0^1\left[\int_0^y xy\,dx+\int_y^1yx\,dx\right]dy\\ =\int_0^1\int_0^1xy\,dx\,dy=\frac14.$$
Remains to compute the standard deviations.
I used a dirty way. I used some programming language like Matlab to generate independent uniform variables and then computed the max, min between them. The results indicated that the correlation is 0.5. However computing it formally is still questionable. – Lan Trần Thị Nov 13 '17 at 10:52