Would you include more specifics about what you want to do. For example, are all of the $X_m$ from the same distribution? Are you interested in the distribution of the sums or just the ability to determine moments of the sum? Is it for the moments of a general $N$ (which means you would also need to make something specific about the individual distributions - as some distributions don't have finite moments).

For example, suppose all distributions were ChiSquare distributions with a common parameter $\nu$:

dist[i_] := ChiSquareDistribution[\[Nu]]
Sum[a[i] Mean[dist[i]], {i, 1, n}]

$$\sum _{i=1}^n \nu a(i)$$

I think you're not using Expectation correctly. I could be wrong though, but from what I see in the documentation, your second argument is not of the required types. It's supposed to be basically x \[Distributed] dist where x is a symbolic variable, not a table of symbolic values.

Edit. My bad, the syntax {x,y,z,...}\[Distributed] dist exists but it's for multivariate distributions. Maybe the syntax {Subscript[x, 1] \[Distributed] Subscript[dist, 1], Subscript[x, 2] \[Distributed] Subscript[dist, 2], \[Ellipsis]} is what you want, but I'm not sure how you would write this with a formal number of parameters.

Edit#2. That being said even X[x, imax] can't be called without imax being defined. Table[x[i], {i,imax}] would return the error 'Table: iterator {i,imax} does not have appropriate bounds'.