Question: Assume independence, and let p ij = n ij /n and ÏÌ ij = p i+ p +j . a. Show that p ij and

Assume independence, and let pij= nij/n and π̂ij= pi+p+j.

a. Show that pij and π̂ij are unbiased for πij = πi+ π+j.

b. Show that var(pij) = Ï€i+ Ï€+j(1 €“ Ï€i+ Ï€+j)/n 

c. Using E(pi+ p+j)2 = E(p2i+) E(p2+j) and E(p2i+) = var (pi+) + [E(pi+)]2, show that

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

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

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