A experiment produces random vector x

Question

An experiment produces random vector X = [X₁ ··· X_k]' with expected value µX = [µ₁ ··· µ_k]' The ith component of X has variance Var[X_i] = σ²_i . To estimate µX, we perform n independent trials such that X(i) is the sample of X on trial i, and we form the vector mean

(a) Show M(n) is unbiased by showing E[M(n)] = µ_X.

(b) Show that the sequence of estimates M_n is consistent by showing that for any constant c > 0,