# Solved – How to make sense out of integration over discrete data points

Looking for a proof of the expected value of the score function equating zero, I came to this document that was recommended in another answer.

Considering that we have a sample of $$n$$ $$x_i$$ values, I cannot figure out why the expected value turns out into an integral instead of a summation: what is the curve of which we're taking the area below it? In my mind I can just see some specific points in a graph, with no area under it, since we have a finite and discrete number of datapoints.

I do understand that the integral is crucial for the proof as to be interchanged with the derivative and then using the pdf probabilities for equating it to 1. But I would not know how to apply all this into a pmf or discrete case.

$$X_i$$ is continuous a random variable, with pdf $$f_{X_i}(x_i;theta)$$, and the expectation requires an integral. The integral limits contain the domain of $$X_i$$. Not $$i$$ from $$1$$ to $$n$$. The $$n$$ samples you have are just realizations of $$X_i$$, i.e. $$X_1,X_2,…,X_n$$. You're not integrating/summing across these variables. You're integrating for a particular $$i$$, let's say $$X_2$$, and obtain an expression for the expected value of interest.