Densities and Cumulative Distribution Functions Has anyone seen this notation before? What does it mean?
$\int_{0}^{\infty} f(x) G(x) dx$
$f(x)$ is a density and $G(x)$ is a cumulative distribution function
 A: I assume both the density $f(x)$ and the cdf $G(x)$ refers to random variables on the interval $(0, \infty)$. Let $X$ be a random variable with the density $f$ and $Y$ be a random variable with the cdf $G $. Then 
$$ \DeclareMathOperator{\E}{\mathbb{E}}  \DeclareMathOperator{\P}{\mathbb{P}}
  \int_0^\infty f(x) G(x) \; dx = \E_X G(X) = \E_X \P (Y \le X)
$$
If you could tell us some more about the context where you saw this, we might tell you why it could be useful.
A: Suppose that $X$ and $Y$ are independent continuous random variables with pdfs $f$ and $g$ respectively, and that
$G$ is the CDF of $Y$. Suppose also that $X \geq 0$.
Then,
\begin{align}
P\{Y \leq X\} & = \int_{-\infty}^\infty \int_{-\infty}^x f(x)g(y)\,
\mathrm dy\,\mathrm dx\\
&= \int_{-\infty}^\infty f(x) \left[\int_{-\infty}^x g(y)\,
\mathrm dy\right]\,\mathrm dx\\
&= \int_{-\infty}^\infty f(x) G(x)\,\mathrm dx\\
&= \int_0^\infty f(x) G(x)\,\mathrm dx
&\scriptstyle{\text{since}~ f(x) = 0~\text{when}~ x < 0}
\end{align}
Note that it is not necessary to assume that $Y$ is
positive (having $X \geq 0$ suffices), and that we
get the expected value of $P\{Y \leq X\}$
(as in kjetil b halvorsen's answer) only in the sense that $P\{Y \leq X\}$ is a constant whose
expected value is just that constant.
If $X$ can take on both positive and negative values, then
$$\int_0^\infty f(x) G(x)\,\mathrm dx 
= P\left(\{Y \leq X\}\cap \{X \geq 0\}\right).$$
