statistics and biased estimator of normal distributions Let $X_1,X_2,X_3,X_4$ be independent, identically distributed random variables from a population with mean $\mu =10$ and variance $\sigma ^2=10$. Let$$\overline Y=\frac{X_1+X_2+X_3+X_4}{4}$$denote the average of these four random variables (in other words, the sample mean).
a) What is the expected value and variance of $\overline Y$?
b) Now, consider a different estimator of $\mu$:$$\displaystyle W=\frac{X_1+X_2+X_3+X_4}{8}.$$This is an example of a weighted average of the $X_i$.  What is the expected value and  variance of $W$?
c) Based on your answers on parts (a) and (b), which estimator of $\mu$ do you prefer, $\overline Y$ or $W$?
Attempt:
a) I got $\mathbb{E}(X)=10$, but variance known formula is $\mathbb{E}(X^2)-\mathbb{E}(X)^2$ but got stuck there.
b) $\dfrac{20}{8}+\dfrac{10}{4}+5=\text{mean}$, and stuck on variance.
c) Checking unbiasness?
 A: a) Your answer for expected value is correct. To find the variance you need to use the formulae:

*

*$\operatorname{Var}(aX)=a^2\operatorname{Var}(X)$ for any real number $a$ and random variable X


*$\operatorname{Var}(X+Y) = \operatorname{Var}(X) +\operatorname{Var}(Y)$ for independent random variables $X$ and $Y$
Therefore,
\begin{align}
\operatorname{Var}\left(\frac{1}{4}\left(X_1 + X_2 + X_3 + X_4 \right) \right) &= \left(\frac{1}{4}\right)^2 \operatorname{Var}\left(X_1 + X_2 + X_3 + X_4\right) \\
 &= \frac{1}{16} \left(\operatorname{Var}(X_1) + \operatorname{Var}(X_2) + \operatorname{Var}(X_3) + \operatorname{Var}(X_4) \right) \\
 &= \frac{1}{16} (10 + 10 + 10 + 10) \\
 &= 2.5
\end{align}
b) Following the method from part a) should give you an answer to this. I'll leave it to you as you will understand it better if you try it yourself!
c) An estimator is unbiased if its expectation is equal to the true mean (in this case, 10). Unbiasedness is generally a good thing. Another thing to consider is the variance of an estimator, which you want to be small (you don't want your estimator to change each time you resample). If both estimators are unbiased, then the better estimator is the one with smaller variance.
