If $f(x)$ is a density function and $F(x)$ is a distribution function of a random variable $X$ then I understand that the expectation of x is often written as:
$$E(X) = \int x f(x) dx$$
where the bounds of integration are implicitly $-\infty$ and $\infty$. The idea of multiplying x by the the probability of x and summing makes sense in the discrete case, and it's easy to see how it generalises to the continuous case. However, in Larry Wasserman's book All of Statistics he writes the expectation as follows:
$$E(X) = \int x dF(x)$$
I guess my calculus is a bit rusty, in that I'm not that familiar with the idea of integrating over functions of $x$ rather than just $x$.
- What does it mean to integrate over the distribution function?
- Is there an analogous process to repeated summing in the discrete case?
- Is there a visual analogy?
UPDATE: I just found the following extract from Wasserman's book (p.47):
The notation $\int x d F(x)$ deserves some comment. We use it merely as a convenient unifying notation so that we don't have to write $\sum_x x f(x)$ for discrete random variables and $\int x f(x) dx$ for continuous random variables, but you should be aware that $\int x d F(x)$ has a precise meaning that is discussed in a real analysis course.
Thus, I would be interested in any insights that could be shared about what is the precise meaning that would be discussed in a real analysis course?