Continuous Random Variables Over Infinite Domains

So far, we've worked with continuous random variables on a bounded interval $[a,b]$, where the probability density function (PDF) $f(x)$ satisfies $\int_a^b f(x)\,dx = 1$. We now extend this idea to infinite domains such as $[0,\infty)$, $(-\infty,0]$, or $(-\infty,\infty)$.

If $X$ takes values on an unbounded interval, its PDF $f(x)$ must still satisfy the normalization condition

$$\int_{-\infty}^{\infty} f(x)\,dx = 1,$$

where any unbounded part of the integral is treated as an improper integral, evaluated via a limit:

$$\int_0^\infty g(x)\,dx = \lim_{b\to\infty} \int_0^b g(x)\,dx.$$

For example, consider $f(x) = 2e^{-2x}$ on $[0,\infty)$. We check normalization:

$$\begin{align*} \int_0^\infty 2e^{-2x}\,dx &= \lim_{b\to\infty}\left[-e^{-2x}\right]_0^b \\ &= \lim_{b\to\infty}\left(1 - e^{-2b}\right) \\ &= 1 - 0 \\ &= 1. \end{align*}$$

Since the integral equals $1$, $f(x) = 2e^{-2x}$ is a valid PDF on $[0,\infty)$.