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albega
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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=1nPXj(Xj)dXj (I really doubt this is right).
Then
<Y>=∫(∑i=1nXi)∏j=1nPXj(Xj)dXj
=∑i=1n∫Xi∏j=1nPXj(Xj)dXj
then all the integrals apart from the ith one go to one because the various probability functions are normalised so
=∑i=1n∫XiPXi(Xi)dXi
=∑i=1n<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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