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.