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