If $X_i\sim B(1, p_i), i=1, ..., n,$ all independent. Let $Q=\min(X_1,\ldots, X_n)$ and $W = \max(X_1, \ldots, X_n).$ Find the correlation between $Q$ and $W.$
Computing correlation by the usual definition $$ \frac{Cov(Q,W)}{\sqrt{\text{Var}(Q) \text{Var}(W)}}$$ is always a very long and tedious approach. Is there any shortcuts to doing this without using the definition?
For example if I consider a bivariate case where $q_i = 1-p_i.$ Then
$E(Q)=E(\min(X_1,X_2)) = p_1, \text{Var}(Q) = p_1 q_1$,
$E(W)=E(\max(X_1,X_2))= q_1, \text{Var}(W) = p_1 q_1,$ so that
$$ Cov(QW) = E(QW)-E(Q)E(W)= E(X_1)E(X_2) - E(Q) E(W) = p_1q_1- p_1q_1 (p_1q_1) = p_1q_1- (p_1q_1)^2 $$ and
$$\sqrt{Var(Q) Var(W)} = \sqrt{(p_1q_1) (p_2q_2)} = p_1q_1.$$
If the results of this bivariate case is correct, how do I extend it in the mulvariate case with regard to my posted question for all $i= 1, \ldots, n?$ It seems to be $Cov(QW) = p_1q_1 - (p_1q_1)^n$ and $\sqrt{Var(QW)} = p_1q_1$ but I'm not sure of that.
You do need to compute $\operatorname{Cov}(Q,W)$ and $\operatorname{Var}(Q)$ and $\operatorname{Var}(W)$. You can get everything from the joint distribution of $(Q,W)$, which is not difficult to obtain (see below). Once you have the joint distribution for $(Q,W)$, you can deduce the marginal distributions for $Q$ and $W$.
What is the joint distribution for $(Q,W)$? Notice that there are only three possible values for the pair $(Q,W)$: $(0,0)$, $(0,1)$, and $(1,1)$. The first case occurs when every $X_i$ equals zero; the third case occurs when every $X_i$ equals one. Your job is to compute the probabilities of these first and third cases; the middle case follows by subtraction.