Let

Upper Y Subscript 1 Baseline comma ... comma Upper Y Subscript n be i.i.d with mean

mu Subscript Upper Y and variance sigma Subscript Upper Y Superscript 2.

Upper Y overbar is an unbiased estimator of mu Subscript Upper Y.

Is Upper Y overbar squared an unbiased estimator of

mu Subscript Upper Y Superscript 2 ?First, note that for a random variable X

var left parenthesis Upper X right parenthesis equals Upper E left parenthesis Upper X squared right parenthesis minus Upper E left parenthesis Upper X right parenthesis squared equals Upper E left parenthesis Upper X squared right parenthesis minus mu Subscript Upper X Superscript 2 and Upper E left parenthesis Upper X Subscript i Baseline Upper X Subscript j right parenthesis equals mu Subscript Upper X Superscript 2 for i not equals j.So,

Upper E left parenthesis Upper Y overbar squared right parenthesis equals Upper E left parenthesis StartFraction 1 Over n EndFraction Summation from i equals 1 to n Upper Y Subscript i right parenthesis squared equals StartFraction 1 Over n squared EndFraction Summation from i equals 1 to n Upper E left parenthesis Upper Y Subscript i Superscript 2 Baseline right parenthesis plus StartFraction 1 Over n squared EndFraction Summation from i equals 1 to n Summation from j not equals i to nothing Upper E left parenthesis Upper Y Subscript i Baseline Upper Y Subscript j right parenthesis

=

mu Subscript Upper Y

mu Subscript Upper Y Superscript 2

sigma Subscript Upper Y Superscript 2

+

StartFraction 1 Over n EndFraction

StartFraction 1 Over n squared EndFraction

StartFraction 1 Over n cubed EndFraction

mu Subscript Upper Y

mu Subscript Upper Y Superscript 2

sigma Subscript Upper Y Superscript 2

Hence,

A.

Yes,

Upper Y overbar squared

is an unbiased estimator of

mu Subscript Upper Y Superscript 2

.

B.

No,

Upper Y overbar squared

is not an unbiased estimator of

mu Subscript Upper Y Superscript 2

.

Upper Y overbar

is a consistent estimator of

mu Subscript Upper Y

.

Is

Upper Y overbar squared

a consistent estimator of

mu Subscript Upper Y Superscript 2

?

Upper E left parenthesis Upper Y overbar squared right parenthesis equals Upper E left parenthesis StartFraction 1 Over n EndFraction Summation from i equals 1 to n Upper Y Subscript i right parenthesis squared equals StartFraction 1 Over n squared EndFraction Summation from i equals 1 to n Upper E left parenthesis Upper Y Subscript i Superscript 2 Baseline right parenthesis plus StartFraction 1 Over n squared EndFraction Summation from i equals 1 to n Summation from j not equals i to nothing Upper E left parenthesis Upper Y Subscript i Baseline Upper Y Subscript j right parenthesis equals mu Subscript Upper Y Superscript 2 Baseline plus StartFraction 1 Over n EndFraction sigma Subscript Upper Y Superscript 2

Hence,

A.

Yes,

Upper Y overbar squared

is a consistent estimator of

mu Subscript Upper Y Superscript 2

.

B.

No,

Upper Y overbar squared

is not a consistent estimator of

mu Subscript Upper Y Superscript 2