# Why is $\mathrm E( x_i)= \mu$ where $\mu$ is the mean of the population when sampling is done without replacement?

$\bullet$ Prove: $\mathrm E(\bar x)= \mu$

Let $x_1,x_2,x_3\ldots,x_n$ denote the sample observations. The sample mean is $$\bar x= \frac{(x_1+x_2+x_3+\ldots+x_n)}{n}= \frac{1}{n}\sum x_i$$ where $x_i$ is the $i$-th member of of the sample.

Note, in simple random sampling(with or without replacement), the sample members has the same probability distribution as in the variable $x$ in the population.

Therefore, $\mathrm E( x_i)= \mu$

And $$\mathrm E(\bar x)= \frac{1}{n}[\mathrm E(x_1)+ \mathrm E(x_2)+\ldots+\mathrm E(x_n)]= \mu.$$

What I'm not getting is the blocked part that the author wanted to highlight.

Can anyone tell me why actually $x_i$ has the same probability distribution as $x$ in the population especially even when the random sampling is done without replacement?

For $x_i$ and $x_j$ are not independent any-more when the sampling is done without replacement; so can then also the probability distribution of $x_i$ and $x_j$ remain the same as that of $x$ in the population?

How can $\mathrm E(x_i)= \mu$ when the sampling is done without replacement?

• Imagine the population is a deck of cards; the value $1$ is assigned to an ace, the value $0$ to any other card; so $\mu=4/52=1/13.$ Now let $n=52,$ i.e., sample the whole deck without replacement. Let $X_i$ be the value of the $i$-th card. Do you believe that the expectations $E(X_1),E(X_2),\dots,E(X_n)$ are all equal? If not, which one do you think is the biggest? In other words: Suppose the whole deck is dealt out, face down; you want to draw an ace, and you can take any card you want. Which card would you pick, to maximize your chances of getting an ace? – bof Mar 21 '16 at 9:22
• @bof: Okay, for the first draw, $\mathrm E(X_1)= \mu$ as all the 52 cards are available for selection. But what next? For the second draw, I've not 52 cards; so how can I expect $\mathrm E(X_2)= \mu\;?$ This is what I'm not understanding. – user142971 Mar 21 '16 at 9:27