Here are the specifics:

1) Yes, all the X[i] should follow the same distribution, as the problem is related to the equiprobable sampling of a (very) big population of solid particles.

2) The list of operations I would wish to do is the following:

• i) the sum of particles N[i] in a batch of any broken material. N[i] is a random variable because they can be randomly selected from the big set.

    Ntot = Sum [N[i]]

Question: how many particles are collected in the sample, if the sampling follows some simple distribution (like the binomial one)? Answer in terms of mean and variance.

• ii) the weighted sum (2 nd step): the total weight of particles in a sample

    W = Sum [w[i]*N[i]]

Question: the same as above, now given as the average weight collected in the sample (and variance)

• iii) a ratio of 2 weighted sums (2 nd step), where ai is the total content of of a certain chemical or physical component (i.e. w[i] is the total weight of each particle, A[i] is just the weight of the component of interest)

  a = Sum [A[i]*N[i]] / Sum [w[i]*N[i]]
  

Question: this is the mean and variance of the "grade" a of the component A in the sample, i.e:

       mean[a] = mean [ A/ W]  
       variance [a] = variance [A/ W]

   Where : 

       A = Sum [A[i]  N[i]]   &&  W = Sum [w[i] N[i]]

The ratio of 2 sums is expected to be a difficult calculation, but just the one sum should be simple, yet I am stuck in the first step.

Thank you.

Just replace the X[i] [Distributed] dist with the pattern:

"wrong definition"->  Xi[x_,imax_]:=Table[x[i], {i, imax}] \[Distributed] dist

"new definition"-> distXi[x_,imax_]:=Table[x[i] \[Distributed] dist, {i, imax}]

Now the example is running OK if I assign a value to imax, for example, imax=3.

However, now I see that the problem is in running sums, not expectations: the result can turn very "ugly" if imax is a very big number.

It is a way to evaluate sums without explicit, numeric iterators? I don't want to have kilometric expressions for big imax values, specially for more complicated distributions or for more complicated functions than expectations, like variances.