You might consider asking this question at CrossValidated to obtain the estimation formulas and then back here for implementation with Mathematica.

You also seem to be assuming subject matter knowledge that not everyone will have. For example, you use "weights" that are actual weights (tons? pounds?) of "broken material". What is "broken material"? There's also mention of "grade a of component A".

It also seems you want to estimate parameters from a sample as opposed to determining parameters from a specified set of distributions (which would involve functions like Expectation or Moment).

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'.