I'm struggling to understand a line of the following proof:
Claim: The sample mean $\bar{X}$ is an unbiased estimator of the population mean $\mu$
Proof
The random variables each based on a single observation $X_1, X_2, …, X_n$ are associated with random samples of size $n$ taken from the population. Then; for each valid $i$, $X_i$ has the same probability distribution as the population.
$\implies E[X_i]= \mu$, thus:
$E[\bar{X}]=E[\frac{1}{n}\sum_{i=1}^{n}{X_i}]=\frac{1}{n}\sum_{i=1}^{n}{E[X_i]}=\frac{1}{n}(n\mu)=\mu$
QED
The issue is I don't know how $E[X_i]= \mu$. How can the expectation of a single observation be the mean of the population? Surely: $E[X_i]=x_i$
Best Answer
Note that each $X_i$ is a random variable and so it's endowed with a distribution. It is specified that the distribution of $X_i$ is such that $EX_i=\mu$. You wonder -- why? Furthermore, it may be confusing that each $X_i$ will always have some specific value $x_i$. Now, important point to realize here is that the association between random variable $X_i$ and its possible values is made according to the principle of simple random sampling, which says that any individual from the population has an equal chance of being selected (sampled). Therefore, Each $X_i$ is associated with an individual (can be any, equally likely) from the population. That population is characterized with a distribution with mean $\mu$. Hence, $X_i$ is characterized by the same distribution. In particular, although a single value of $EX_i$ is $x_i$, its average or expectation $EX_i=\mu$ (not $x_i$).