Let X1 and X2 be independent, each with unknown mean µ and known variance =1.

a. Let Find the bias, variance, and mean squared error of

b. Let Find the bias, variance, and mean squared error of

c. Let Find the bias, variance, and mean squared error of

d. For what values of does have smaller mean squared error than?

e. For what values of n does have smaller mean squared error than

Answer:

Step 1 of 5:

Given, let and be independent, with unknown mean and known variance = 1.

Let, = then the claim is to find the bias, variance and mean squared error (MSE) ofBias of = E( ) -

We know that mean of is E( ) and variance of is V( )

E( ) =

=

=

=

Hence, mean of is

The bias of = E( ) -

= -

= 0

Therefore, The bias of is 0.

variance of is V( )

V( ) =

=

=

=

We have, = 1

Therefore, V( ) =

The mean square error of = V( ) + (bias

= + 0

=

Hence, the mean square error of is

Step 2 of 5:

Let, = then the claim is to find the bias, variance and mean squared error (MSE) ofBias of = E( ) -

We know that mean of is E( ) and variance of is V( )

E( ) =

=

=

=

Hence, mean of is

The bias of = E( ) -

= -

= 0

Therefore, The bias of is 0.

variance of is V( )

V( ) =

=

=

=

We have, =

Therefore, V( ) =

The mean square error of = V( ) + (bias

= + 0

=

Hence, the mean square error of is