Bias and MSE

Question

We estimate the unknown mean θ of a random variable X with unit variance by forming the sample mean Mn=(X1+⋯+Xn)/n of n i.i.d. samples Xi and then forming the estimator

Θˆn=13⋅Mn.

Your answers below can be functions of θ and n. Follow standard notation and use 'theta' to indicate θ.

The bias E[Θˆn]−θ of this estimator is:
unanswered

The mean squared error of this estimator is:
unanswered

Answer 1

For biais: -2*theta/3

Answer 2

-2*theta/3

Answer 3

mse: 4*theta^2/9

Answer 4

Helper's answer is kinda right, but input it like:
(4*theta^2)/9

Answer 5

Actually above is just part of the answer, mse is 1/9+(4*theta^2)/9

Answer 6

typo above: (1/9*n)+(4*theta^2)/9

Answer 7

(1/(9*n))+(4*theta^2)/9

Answer 8

-2*theta/3

Answer 9

bias: -2*theta/3

mse: (1/(9*n))+(4*theta^2)/9