Think of a simple random sample X1,...,X5 following the distribution N(mu,sigma^2). Show that the statistic W=(X1+4X4-2X5)/3 is an unbiased estimator of mu.

Question

Also: What distribution does W follow? Specify the mean and variance.

Answer 1

An estimator θˆ is called unbiased if it expectation is equal to the true value of the parameter.

E[W]=E[(X1+4X4-2X5)/3]=(1/3)(E[X1]+4E[X4]-2E[X5])=

=(1/3)(μ+4μ-2μ)=μ, So E[W]=μ which mean that W is unbiased estimator of μ.

Linear combination of normal random variables again is Normal random variable. So W follows Normal Distribution

Mean would be sum of means: (1/3)(μ+4μ-2μ)=μ; Variance (7 Sigma^2)/3.

If you'd like to find dsitribution of W itself, the characteristic function's technic would be the best.