Chapter 7 Common Distributions of Continuous Random Variables

A continuous random variable can take any value within some uncountable interval, such as $[0, 1]$, $[0,\infty)$, or $(-\infty, \infty)$. These are often measurement type variables.

The probability density function (pdf) (a.k.a., density) of a continuous random variable $X$, defined on a probability space with probability measure $\textrm{P}$, is a function $f_X:\mathbb{R}\mapsto[0,\infty)$ which satisfies $$\begin{align*} \textrm{P}(a \le X \le b) & =\int_a^b f_X(x) dx, \qquad \text{for all } -\infty \le a \le b \le \infty \end{align*}$$

For a continuous random variable $X$ with pdf $f_X$, the probability that $X$ takes a value in the interval $[a, b]$ is the area under the pdf over the region $[a,b]$.

The axioms of probability imply that a valid pdf must satisfy $$\begin{align*} f_X(x) & \ge 0 \qquad \text{for all } x,\\ \int_{-\infty}^\infty f_X(x) dx & = 1 \end{align*}$$

In this section we study some commonly used continuous distributions and their properties. When developing a probability model for a random process, certain assumptions are made about the process or the distribution of a corresponding random variable. Some situations are so common that the corresponding distributions have special names.