Probability

Bias and MSE

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:

The mean squared error of this estimator is:

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1. For biais: -2*theta/3

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2. -2*theta/3

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3. mse: 4*theta^2/9

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

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

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

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7. (1/(9*n))+(4*theta^2)/9

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8. -2*theta/3

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