Suppose X_1, X_2 and X_3 are sampled independently where E (X_i) = mu for all i.

Question

Suppose X_1, X_2 and X_3 are sampled independently where E (X_i) = mu for all i. Suppose X_1, X_2 and X_3 come from populations with variances 4 sigma_2, 2 sigma^2 and 5 sigma^2. Consider the point estimate mu for the populations' mean mu: mu = aX_1 + bX_2 + cX_3 whore a, b and c are constants. (a) Determine what values a, b, c can have such that mu is an unbiased estimator of mu (i.e. what equation (s) should a, b and c satisfy to ensure mu is unbiased. (b) Given an unbiased mu in the form above where the variance is minimized. (c) Given the observed sample {x_1, x_2, x_3} = {10, 3, 5}, estimate mu. In our first class, we collected random samples {X_, ..., X_n} uniformly at random between 0 and c. Some of you estimated c with c = max {X_1, ..., X_n} and we mentioned that this is a biased (but good) estimator of c. In this problem we find the bias of c. (a) Suppose u is some constant between 0 and c. What is the probability that X_i is at most u? (b) What is the probability that alt X_i are at most u? (c) Let c = max {X_1, ..., X_n} be our estimate for c. What is the PDF for c? (d) What Is E (c)? (e) What is bias (c)? What happens to the bias as n grows large?

Explanation / Answer

Given that X1, X2 and X3 are independent where E(Xi) =  for all i and variances are 42 , 22 and 52.

COnsider the point estimate µ^ for the population mean is aX1+bX2+cX3

a) Unbiasedness :

An estimator is said to be unbiased iff,

E(µ^) = µ

E(µ^) = E(aX1+bX2+cX3) = aE(X1) + bE(X2) + cE(X3)

=aµ + bµ + cµ = µ(a+b+c)

Now we have given that, (a,b,c) = (10,3,5) we have to find µ.

E(µ^) = µ(a+b+c) = (10+3+5)µ = 18µ

Here this estimator is biased because E(µ^) not = µ.

For unbiased estimator we need set of a,b,c is :

1 0 0

0 1 0

0 0 1

For these three value of a,b,c the estimator is unbiased.

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