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Let Y, Y2, Yz and Y4 be independent, identically distributed random variables from a population with mean u and variance o. L

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Answer #1

Here Y_i has mean \mu and variance \sigma ^2 .

Now the sample mean is (Y1 + Y2 + Y3 +Y4)

i) The expected value of \overline{Y} is

(Y1) + E (Y2) + E(Y3) + E(Y4)) E (T) = E () = 11

The variance of \overline{Y} is

(Var (Y1) + Var (Y2) + Var (Y3) + Var (Y4)) var (7) - 16 var (7) =ๆ\

ii) Consider the estimator is 00- 以 + + 111 + ト

The expected value of W is

E(W) = E(VL) + F(X2) + E() + (Y) E (W) = x + + E(W) =

Hence 00- 以 + + 111 + ト is an unbiased estimator of \mu.

The variance is

Var (W) = az Var (Y4) + ez Var (Y2) + dvar (Y3) + zz Var (Y4) Var (W) = 110?

Since the variance of (Y1 + Y2 + Y3 +Y4) is lesser \left (\frac{\sigma ^2}{4} <\frac{11\sigma ^2}{32} \right ) than that of 00- 以 + + 111 + ト

we prefer the estimator \overline{Y} .

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