Assume independence, and let p_ij= n_ij/n and π̂_ij= p_i+p_+j.

a. Show that p_ij and π̂_ij are unbiased for π_ij = π_i+ π_+j.

b. Show that var(p_ij) = Ï€_i+ Ï€_+j(1 €“ Ï€_i+ Ï€_+j)/n 

c. Using E(p_i+ p_+j)² = E(p²_i+) E(p²_+j) and E(p²_i+) = var (p_i+) + [E(p_i+)]², show that

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) = {T;+ T+[T;+ (1 – 7+;) + T +; (1 – T;+ var( î¡;) + π- (1- π.) π., (1- π.;)/n'.