If Y=X1+X2+...+XN prove that <Y>=<X1>+<X2>+...+<XN>
<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 all the integrals apart from the ith one go to one because the various probability functions are normalised so
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...