# 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.