Just as the difference between two sample means is normallydistributed for large

Question

Just as the difference between two sample means is normallydistributed for large samples, so is the difference between twosample proportions. That is if Y1 and Y2 are independent binomialrandom variables with parameters (n1 , p1) and (n2, p2),respectively, then (Y1 / n1) - (Y2 / n2 ) is approximatelynormally distributed for large values of n1 and n2. Find E( Y1/ n1 -   Y2 / n2 ) Just as the difference between two sample means is normallydistributed for large samples, so is the difference between twosample proportions. That is if Y1 and Y2 are independent binomialrandom variables with parameters (n1 , p1) and (n2, p2),respectively, then (Y1 / n1) - (Y2 / n2 ) is approximatelynormally distributed for large values of n1 and n2. Find E( Y1/ n1 -   Y2 / n2 )

Explanation / Answer

given that Y1~ B(n1,p1)=> E(Y1) = n1 P1 andV(Y1) =  n1 P1Q1 and        Y2 ~ B(n2,p2)=>  E(Y2) = n2 P2 andV(Y2) =  n2 P2Q2 E(Y1/n1 -Y2/n2) = E(Y1/n1) - E(Y2/n2 ) (addition theorem ofexpectations)                              = (1/n1 )E(Y1) -(1/n2 )E(Y2)                            =  (1/n1 )n1P1 -(1/n2 ) n2P2             E(Y1/n1 -Y2/n2)  =  P1 -   P2

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