I need some help with this question.
Suppose I am given $2n-1$ random variables $X_1$, $X_2$ ... , where $\Bbb E(X_i)=\mu$ and $\DeclareMathOperator{\var}{var}\var(X_i)=\sigma^2$ for $i=1,2...2n-1$. Then let $Y = X_1 + X_2 + ... + X_n$ and $W = X_n + X_{n+1} + ... X_{2n-1}$. Now I am asked to calculate $\DeclareMathOperator{\cov}{cov}\cov(Y,W)$.
I tried solving this by expanding \begin{align} \cov(Y,W) &= \cov(X_1 + X_2 + ... + X_n, X_n + X_{n+1} + ... + X_{2n-1})\\& = \cov(X_1, X_n) + \cov(X_1, X_{n+1}) + ... + \cov(X_n, X_{2n-1}), \end{align} such that there are $n^2$ covariance terms, then I am stuck. Since I cannot assume that the variables are independent, how to I calculate the covariance of every single pair of variables? Since for instance, $\cov(X_1, X_n) = \Bbb E(X_1X_n) - \Bbb E(X_1)\Bbb E(X_n)$, then $\Bbb E(X_1)=\Bbb E(X_n)=\mu$ but how to I get the value of $\Bbb E(X_1 X_n)$?
Or is there any alternative method to solve this question?