Bias and Variance of Estimators for Population Mean

[gutnedawg]

Homework Statement

Suppose you have a random sample {X1,X2,X3} of size n=3. Denote the population mean $$\mu$$ = E(Xi) and the population variance $$\sigma$$^2= VAR(Xi). Consider the following three estimators for $$\mu$$

$$\mu$$ 1= X
$$\mu$$ 2= X1/5 +X2/2 +X3/5
$$\mu$$ 3= (X1+X2+X3)/5

What is the bias associated with each estimator?
What is the variance associated with each estimator?

Homework Equations

I'm not sure

The Attempt at a Solution

I'm really at a loss now.

 

[xxxx]

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[statdad]

Homework Statement

Suppose you have a random sample {X1,X2,X3} of size n=3. Denote the population mean $$\mu$$ = E(Xi) and the population variance $$\sigma$$^2= VAR(Xi). Consider the following three estimators for $$\mu$$

$$\mu$$ 1= X
$$\mu$$ 2= X1/5 +X2/2 +X3/5
$$\mu$$ 3= (X1+X2+X3)/5

What is the bias associated with each estimator?
What is the variance associated with each estimator?

Homework Equations

I'm not sure

The Attempt at a Solution

I'm really at a loss now.

For each estimator use the rules for expectations and variances of linear combinations of independent variables.

$$\begin{align*} E(aX + bY) &= aE(X) + bE(Y) \\ Var(aX + bY) & = a^2 Var(X) + b^2 Var(Y) \end{align*}$$

Try these with your estimators and show what you find.