Lets say i have a population of size N, where N > 1, with the values $y_{1},y_{2},y_{3},y_{4}, ... , y_{N}$. (Unknown Population Distribution)
Step 1: A Simple Random Sample without Replacement is taken out of size $n_{1}$ from a population of $N$. Let's call it Sample 1 $S_{1}$.
Step 2 : A Simple Random Sample without Replacement is taken out of size $n_{2}$ from a population of $N-n_{1}$. Let's call it Sample 2 $S_{2}$.
I have to show that covariance between the mean of these two samples is
$Cov(\bar y_{S1}, \bar y_{S2}) = \frac{-\sum_{i=1}^{N}(y_{i}-\bar{y})^2}{N(N-1)}$
where $\bar{y}=\frac{1}{N}\sum_{i=1}^{N}y_{i}$
So far what i have is this :-
$Cov(\bar y_{S1}, \bar y_{S2}) = E(\bar y_{S1}\bar y_{S2}) - E(\bar y_{S1})E(\bar y_{S2})$
where $E(\bar y_{S1})=\bar{y}$ which makes $Cov(\bar y_{S1}, \bar y_{S2}) = E(\bar y_{S1}\bar y_{S2}) - (\bar{y}) ^2$
Am unable to proceed any further, not even sure if what i have so far is right.
Any help will be appreciated. Thanks :)