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## Homework Statement

If Y=X1+X2+...+XN prove that <Y>=<X1>+<X2>+...+<XN>

## Homework Equations

<Y>=∫YP(Y)dY over all Y.

## The Attempt at a Solution

I only seem to be able to show this if the Xi are independent, and I also think my proof may be very wrong. I basically have said that we can write the probability in the interval X1+dX1, X2+dX2,..., XN+dXN, as

∏

_{j=1}

^{n}P

_{Xj}(X

_{j})dXj (I really doubt this is right).

Then

<Y>=∫(∑

_{i=1}

^{n}Xi)∏

_{j=1}

^{n}P

_{Xj}(X

_{j})dXj

=∑

_{i=1}

^{n}∫Xi∏

_{j=1}

^{n}P

_{Xj}(X

_{j})dXj

then all the integrals apart from the ith one go to one because the various probability functions are normalised so

=∑

_{i=1}

^{n}∫XiP

_{Xi}(X

_{i})dXi

=∑

_{i=1}

^{n}<Xi>

however in saying all the integrals go to one, I have assumed I could separate all the integrals, i.e that the variables were independent.

Also, is there not a really easy way to prove this - I can't seem to find any books/websites proving it making me think it's just trivial...

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