what is the expectation value of a subset of random variables?

Question

suppose $x_1, x_2, ..., x_{100}$ are identical but correlated random variables. I want to know if there is any relationship between $\langle |x_1+x_2+...+x_{100}|\rangle$ and $\langle |x_{25}+x_{26}+...+x_{75}|\rangle$? (are they same?!). where $\langle... \rangle$ denotes expectation value.

Answer 1

For random variables $X$, $Y$ and $Z=X+Y$ $$E[Z]=E[X]+E[Y]$$ that is, the expected value is linear. So since each variable is identical, $$\forall n(E[x_n]=E[x_{n+1}]=E[x_{n+1}]...)$$ $$\therefore E[x_n+x_{n+1}...+x_{n+m}] = mE[x_n] $$So $$⟨x_1+x_2+...+x_{100}⟩=2⟨x_{25}+x_{26}+...+x_{75}⟩$$ simply because there are twice as many random variables in the first sum