For a random sample of size n from a population of size N,

Chapter , Problem 24

(choose chapter or problem)

For a random sample of size n from a population of size N, consider the following as an estimate of \(\mu\):

                                             \(\bar{X}_{c}=\sum_{i=1}^{n} c_{i} X_{i}\) 

where the \(c_i\) are fixed numbers and \(X_{1}, \ldots, X_{n}\) is the sample.

a. Find a condition on the \(c_i\) such that the estimate is unbiased.

b. Show that the choice of \(c_i\) that minimizes the variances of the estimate subject to this condition is \(c_{i}=1 / n\), where i = 1, . . . , n.