Densities and Cumulative Distribution Functions

Question

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

Answer 1

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 $$\int_0^\infty f(x) G(x)\,\mathrm dx = P\left(\{Y \leq X\}\cap \{X \geq 0\}\right).$$

Answer 2

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.