By definition the expected value of an is...
Marry Christmas from Serbia
To be more precise on what chisigma wrote above, you'd be looking for the expected value of (in general, capital letters point to random variables, whereas small letters point to variables).
But here, it's much better to calculate the first 2 moments of , then say that since the joint pdf can be written as functions of only x1, x2 or x3, the three variables are independent. And hence this simplifies quite a lot the calculations, especially for the last 2 expectations.
You'd be able to treat the other rv's as constants if you're considering a conditional expectation with respect to some given rv or sigma-algebra. But given the kind of problem you're working on, it's not something you've studied yet, maybe soon
Just an update.
I am using chisigmas method for this question as a means of practice..
for the first one E[3X1+5X2+4X3] is there a shorthand alternative other than calculating the first 2 moments? As opposed to integrating:
E[3X1+5X2+4X3] * (X1X2X3)/3
the density of X3 by inspection, no integration, is on (0,1)
that's a beta distribution, the mean is alpha/(apha+beta)=2/3
you can do the same with the other two random variables, even though they aren't beta's