What is is it meant by "analytically derive the expected value of an estimator?" in statistics? Good evening, everyone. I am new to statistics, so I wanted to see if I could get guidance with this problem:
Consider a variable Y which follows a distribution with mean µ and variance σ2 in the population.
Suppose we take a random sample of Y of size n, and propose to estimate the mean of Y from this
dataset using two potential new estimators, the Addition Estimator and the First-ten Estimator, which
are defined as follows:
The Addition Estimator: Take the sample mean, and add on $\frac{20}{n}$
The First-ten Estimator: Select only the first ten observations in the sample, and take the sample
mean of these observations
For the sake of simplicity, we will assume that we always take samples containing more than ten obser-
vations. Remember that the sample mean can be calculated in the following manner:  $µ = ∑_n \frac{Yn}{n} $
Analytically derive the expected value of both estimators. Based on these results, is each estimator
biased or unbiased? Using your knowledge of limits, intuitively explain what you think will happen
to the bias (that is, the degree to which E[θ] deviates from θ) as n approaches infinity
I am not understanding what it means to "analitically derive de expected value" if someone could explain me what it means and how to do it, I would be very thankful.
Thank you again!
 A: Analytically derive the expected value of the estimator means to take the expected value. Here is the addition estimator: $\hat a=\frac{\sum_{i=1}^n Y_i}{n}-\frac{20}{n}$ and here is the first-ten estimator $\hat b=\frac{\sum_{i=1}^{10} Y_i}{10}$. The question wants you to apply the operation $\mathbb E$ to the addition and first-ten estimators. What is $\mathbb E(\hat a)$ and $\mathbb E (\hat b)$? For the first one, it is $\mathbb E\left(\frac{\sum_{i=1}^n Y_i}{n}-\frac{20}{n}\right)=\frac{\sum_{i=1}^n \mathbb E(Y_i)}{n}-\frac{20}{n}=\frac{n\mu}{n}-\frac{20}{n}=\mu-\frac{20}{n}$. This is biased because we do not expect in the long run, if we take the addition estimators for a lot of samples, that the average estimate will be $\mu$. It will be smaller than $\mu$. But the bias will decrease as $n\rightarrow \infty$ because as n gets larger, 20/n gets smaller. If we take samples of size 10, there will be a larger bias $\mu-\left(\mu-\frac{20}{10}\right)=2$ than if we take samples of size 1000, with bias of $\frac{20}{1000}=.2$. As n goes to infinity, the bias will go to zero. Now try to answer these questions for the first-ten estimator. (1) What does $\mathbb E(\hat b)$ simplify to? (2) Is the estimator biased? (Is the expected value $\mu$?) (3) Does the bias change as $n\rightarrow \infty$, or does it stay the same?
