Just a simple notational question: I want to concisely denote sampling $n$ times from a distribution and summing over a function applied to the samples. I am not sure how to do this however. I cannot define a set of $X_i$ as global variables ahead, because the summation occurs in the scope of an outer summation. And each ieration the inner sum samples anew. Currently, I have
$$\sum_k^N\sum_{x \sim X}^n f(x)$$
But this annoys me. It is abuse of notation and makes no sense, as the end index $n$ has no associated running index like $i = 1$ defined below the summation symbol. Is there an exact way to denote this or can I rely on the reader to understand what this notational hack is supposed to mean from the context, where I talk about the formula?
Best Answer
First use some notation to define your sample. Let $X_1, X_2,\ldots,X_n$ be the $n$ members in your sample. If $f$ is the function you're applying to each member, then the sum you are looking for can be written $ \sum_{i=1}^n f(X_i) $.
If you have a different set of $X$'s within each outer iteration, then you could modify your notation: $X_{1,1}, X_{1,2},\ldots,X_{1,n}$ are the items in iteration 1, then $X_{2,1}, X_{2,2},\ldots,X_{2,n}$ are the items on iteration 2, and so on. The overall sum would then be written $\sum_{k=1}^N\sum_{i=1}^n f(X_{k,i})$.