I am trying to understand how to calculate the expected value of a hypergeometric variable using indicator random variables. The derivation that I read in the book (Introduction to Probability Theory, Hoel Port Stone) is as follows:
Assume the population size to be $r$, of which $r_1$ are of type 1 and $r-r_1$ are of type 2. A sample of size $n$ is drawn without replacement from this population. Let $X_1, X_2, ... X_n$ be indicator random variables where $X_i = 1$ if and only if the ith element in the sample is of type 1. Then,
$E[X_i] = P(X_i = 1) = \frac{r_1}{r}$
I don't understand how the expectation of $X_i$ is the same $\forall i$. Since sampling is done without replacement in hypergeometric distribution, the probability of ith element in the sample being type 1 shouldn't be the same $\forall i$.
Can someone explain why this is true?
Edit: We can write,
$P(X_i=1) = \sum_{x_1}\sum_{x_2}...\sum_{x_{i-1}} P(X_1=x_1, X_2=x_2, ... , X_{i-1} = x_{i-1}, X_i=1)$
where $x_i$'s take values $0$ or $1$.
Can we compute this sum to show $P(X_i=1) = \frac{r_1}{r}?$