# Notation of integral.

In some probability book I've come across this notation: $E(X)=\int_{-\infty}^{\infty}x d F(x)$, and it's very confusing, when I see other books defining the same concept as: $E(X)=\int_{-\infty}^{\infty}xf(x) dx$

I wonder... what does that kind of notation mean?

## 1 Answer

In this case, $f(x)$ would be the probability density function and $F(x)$ would be the cumulative distribution function. They are related by $$f(x)=\frac{dF(x)}{dx}\ ,$$ so $dF(x)=f(x)\,dx$ and the two integrals are equivalent.

In practice, to evaluate the first integral you would probably begin by transforming it to the second anyway.

• Yes, I knew $f(x)$. But I hadn't ever seen the notation $dF(x)$ in an integral just like that. It makes sense like that. – David Molano Mar 2 '15 at 1:50
• One should also mention that there are cases where $F$ is not differentiable. Sometimes, it is then possible to interpret $\int_{-\infty}^\infty x\mathrm{d}F(x)$ as a Riemann–Stieltjes integral. (A common case would be that $F$ is piecewise differentiable, then the integral can be evaluated by considering the pieces individually.) – Eike Schulte Mar 2 '15 at 8:54