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.
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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
Colin Hicks
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my question is about the absolute value. see the question. – mehrd97d Sep 23 '19 at 08:52
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in general, this is not an easy question then as there is no information really about the underlying distribution. The distribution of a sum of random variables is their convolution which will yield wildly different results depending on the the distribution of the variables. The absolute value will then once again dramatically change the distribution. To see what I'm saying about absolute value look at a folded normal distribution which is already quite complex. – Colin Hicks Sep 23 '19 at 16:21