Let be a random sample from an arbitrary and continuous distribution with cumulative distribution function F(x-). Let , , , $\displaystyle P_4 = P(X_1 + X_2 > 0, X_1 + X_3 >0)$. Now let $\displaystyle W_ij = 1$, if $\displaystyle X_i + X_j > 0$ otherwise $\displaystyle W_ij = 0$. For all distinct h, i, j, k
SHOW THAT
$\displaystyle E(W_ii) = p_1$,
$\displaystyle Var(W_ii) = p_1 - (p_1)^2$,
$\displaystyle E(W_ij) = p_2$,
$\displaystyle Var(W_ij) = p_1 - (p_2)^2$,
$\displaystyle Cov(W_ij, W_ik) = p_4 - (p_2)^2$,
$\displaystyle Cov(W_ii, W_ik) = p_3 - (p_1)(p_2)$,
$\displaystyle Cov(W_ij, W_hk) = 0$