Question

(a) Suppose X1, ..., Xn1 i.i.d. ∼ N(µ1, σ2 1 ) and Y1, ...Yn2 i.i.d. ∼ N(µ2, σ2 2 ) are independent Normal samples. Suggest an unbiased estimator for µ1 − µ2 and find its’ standard error. Now Suppose n1 = 100, n2 = 200, it is known that σ 2 1 = σ 2 2 = 1, and we calculate X¯ = 5.7, Y¯ = 5.2. Find a 2-standard-error bound on the error of estimation. (b) Suppose X ∼ Binomial(n1, p1) and Y ∼ Binomial(n2, p2) are independent Binomial random variables. Suggest an unbiased estimator for p1 − p2 and find its’ standard error.

Answer 1

We observe that X1, X2 , ..........Xn1 be a random sample of size n1 is drawn from the normal population with mean 1 and standard deviation 21

and Y1,Y2,........Yn2 be another independent sample another normal population with mean 2 and standard deviation 22

mean of first sample =^X_i//n₁ and mean of second sample y^- = Y_i/n₂

Unbiased estimator of _{1- 2}

₍1 - 2 is an unbised estimator of 1- 2

since E( - y^-- ) = E() - E(y^-- ) =  1- 2

Standard error of [ - y^--] = S.E [ ₍ - y^- ] = V[₍ - y^--] = [₂₁²_/n1 +[₂₂²_/n1]

now suppose n1 =100, n2= 200 = 5.7 y^-- =5.2 and 12 =22= 1

Standard error of difference of sample means =  [₂₁²_/n1 +[₂₂²_/n1] = [1/100 + 1/200 ] =0.12247

1standard error bound is (x^-- y^-) S.E ( x^-- y^-)

0.5 0.12247

(0.37753 ,,0.62247)

2 standard error bound is (x^-- y^-) 2S.E ( x^-- y^-)

0.5 2x0.12247

(0.255 , 0.745 ) similarly calculate 3 standard error bound

2. if X be a variable which follows binomial distribution with parameters n1, P1

and Y be another independent variate follows binomial distribution with parameters n2 , P2

(p1 -p2) is an unbiased estimator of (P1-P2)

Since E (p1 -p2) = E(X/n1 - Y/n2) = E(X)/n1 - E(Y) /n2 = n1 P1 /n1 - n2P2 /n2 = P1- P2

and standard error of  (p1 -p2) = S.E ( p1 - p2 ) = V(p1) + V(p2) = P1Q1/n1 + P2Q2/n2

where Q1 =1-P1 and Q2 = 1-P2

V(p1) = V(X1) / n²₁ = n1P1Q1 / n²_{1 =} P1Q1/n1

similarly V( p2)= P2Q2/n2