# Let X 1 and X 2 be independent, each with unknown mean and known variance

## Question:

Let X_{1} and X_{2} be independent, each with unknown mean μ and known variance σ^{2} = 1.

a. Let μ̂_{1} = X_{1} + X_{2}/2. Find the bias, variance, and mean squared error of μ̂_{1}.

b. Let μ̂_{2} = X_{1} + 2X_{2}/3. Find the bias, variance, and mean squared error of μ̂_{2}.

c. Let μ̂_{3} = X_{1} + X_{2}/4. Find the bias, variance, and mean squared error of μ̂_{3}.

d. For what values of μ does μ̂_{3} have smaller mean squared error than μ̂_{1}?

e. For what values of μ does μ̂_{3} have smaller mean squared error than μ̂_{2}?

Fantastic news! We've Found the answer you've been seeking!

## Step by Step Answer:

**Related Book For**