What does the notation $\int_A$ mean, where $A$ is an event in a probability space?

Question

I am used to seeing integral notation like this, which means the integral over the domain from a to b.

$$ \int_{a}^{b} $$

But now I am looking at a statistics book that says "let A be an event" and then shows the probability of that event like this

$$ \int_{A} fx(X)dx = P(X ∈ A) $$

How do I interpret the notation when only the bottom symbol is given?

Answer 1

You are integrating over the set A. To do an integral, you don't necessarily need to lie in an interval (the points don't even have to be next to each other, you only need that the integral is computable). Intuitively (informally), you only need a set such that the set has "enough" points so that the integral makes sense (if you are interested Lebesgue measure for an introduction to this concept).

So:

$$ \int_{A} fx(X)dx = P(X ∈ A) $$

means just that, integrate the density function over the set A. If you want you can imagine its a summation, and you adding the infinitesimal points according to the density $f_X(x)$ over the elements of $A$.

Answer 2

You integrate over whatever the region is.

As an example, let $X$ be normal. What's $P(X>0 \text{ or } X<-5)$?

$$\int_{-\infty}^{-5}\frac{1}{\sqrt{2\pi}}e^{-x^2/2}dx+\int_0^{\infty}\frac{1}{\sqrt{2\pi}}e^{-x^2/2}dx$$