Definitions for Continuous Random Variables

In the previous chapter we mainly focused on discrete random variables whose set of possible values is either finite or countably infinite. In this chapter, we study random variables whose set of possible values is uncountable. We’ll see later on that a lot of the cases we’ve discussed have analogs in the continuous case.

Probability density function

The continuous random variable is a random variable with infinite possible outcomes (a subset of the real line). We say $X$ is a continuous random variable if there exists a nonnegative function $f$ defined for all $x \in (-\infty, \infty)$, having the property that for any set $B$ of real numbers, $$ P\{x \in B\} = \int_B f(x)dx $$ Here the function $f$ is called the probability density function (PDF). It resembles the probability mass function in the discrete case. The PDF has the following properties:

1. $\int_{-\infty}^\infty f(x)dx = P\{X \in (-\infty, \infty)\} = 1$
2. $P\{a \leq x \leq b\} = \int_a^b f(x)dx$
3. $P\{X = a\} = \int_a^a f(x)dx = 0$

We can also define the cumulative distribution function for a continuous random variable: $$ \begin{aligned} F(a) &= P\{X \leq a\}, \quad B \in (-\infty, a] \\ &= \int_{-\infty}^a f(x)dx \end{aligned} $$

Properties example

These properties often come in handy when we have unknown quantities in a PDF. Suppose $X$ is a continuous random variable with probability density function $$ f(x) = \begin{cases} C\left( 4x - 2x^2 \right), & 0 < x < 2 \\ 0, & \text{otherwise} \end{cases} $$ and we’d like to find $C$ as well as the probability $P\{X > 1\}$. $$ \begin{aligned} 1 &= \int_{-\infty}^\infty f(x)dx \\ &= \int_{-\infty}^0 f(x)dx + \int_0^2 f(x)dx + \int_2^\infty f(x)dx \\ &= C\int_0^2 (4x - 2x^2)dx \\ &= C\left(\int_0^2 4xdx - \int_0^2 2x^2dx \right) \\ &= C \left( 2x^2\bigg|_0^2 - \frac{2}{3}x^3 \bigg|_0^2 \right) \\ &= C\left( 8 - \frac{16}{3} \right) \end{aligned} $$ Now we can easily find $C = \frac{3}{8}$. With the PDF given, it’s trivial to find the CDF:

$$ \begin{aligned} P{X > 1} &= \int_1^\infty f(x)dx \\ &= \int_1^2 \frac{3}{8}\left(4x - 2x^2\right)dx \\ &= \frac{3}{8}\left( 2x^2\bigg|_1^2 - \frac{2}{3}x^3 \bigg|_1^2 \right) \\ &= \frac{3}{8}\left( 8-2 - \left(\frac{16}{3} - \frac{2}{3}\right) \right) = \frac{1}{2} \end{aligned} $$

Lifetime example

Suppose $X$, the lifetime of an item, is a continuous random variable with a density function $$ f(x) = \begin{cases} \lambda e^{-x/100}, & x \geq 0 \\ 0, & x < 0 \end{cases} $$ What is the probability that the item functions between 50 and 150 days?

In mathematical terms, we want to calculate $P\{50 \leq X \leq 150\}$. We first need to find the value of $\lambda$.

$$ \begin{aligned} 1 &= \int_{\infty}^\infty f(x)dx \\ &= \int_0^\infty \lambda e^{-\frac{x}{100}}dx \\ &= \lambda \int_0^\infty e^{-\frac{x}{100}}dx \\ &= -100\lambda \int_0^\infty e^{-\frac{x}{100}} d\frac{-x}{100} \\ &= -100\lambda e^{-\frac{x}{100}} \bigg|_0^\infty \\ &= -100\lambda(0 - 1) = 100\lambda \end{aligned} $$

Recall that $\int e^xdx = e^x$, and $d(ax) = a \cdot dx$ because the derivative is a linear function.

With $\lambda = \frac{1}{100}$, we can calculate

$$ \begin{aligned} P{50 \leq X \leq 150} &= \int_{50}^{150} \frac{1}{100}e^{-\frac{x}{100}}dx \\ &= \frac{-100}{100}\int_{50}^{150}e^{-\frac{x}{100}} d\frac{-x}{100} \\ &= -\left( e^{-\frac{x}{100}}\bigg|_{50}^{150} \right) \\ &= e^{-\frac{1}{2}} - e^{-\frac{3}{2}} \approx 0.383 \end{aligned} $$

Expectation and variance

Earlier we’ve defined the expectation for discrete random variables. If $X$ is a continuous random variable with probability density function $f(x)$, we have $$ f(x)dx \approx P\{x \leq X \leq x + dx\} $$

so it’s easy to find the analog for the expectation of $X$ to be

$$ E[X] = \int_{-\infty}^\infty xf(x)dx $$

Similarly, the expected value of a real-valued function of $X$ is

$$ E[g(x)] = \int_{-\infty}^\infty g(x)f(x)dx $$

which can be used to derive the variance of $X$ $$ Var(X) = E[X^2] - E[X]^2 $$

Suppose $X$ is a continuous random variable with density function $$ f(x) = \begin{cases} 2x, & 0 \leq x \leq 1 \\ 0, & \text{otherwise} \end{cases} $$

$$ \begin{aligned} E[X] &= \int_{-\infty}^\infty xf(x)dx \\ &= \int_0^1 x \cdot 2x dx \\ &= \frac{2}{3}x^3 \bigg|0^1 = \frac{2}{3} \\ E[X^2] &= \int{-\infty}^\infty x^2 f(x)dx \\ &= \int_0^1 2x^3 dx \\ &= \frac{1}{2}x^4 \bigg|_0^1 = \frac{1}{2} \\ Var(X) &= \frac{1}{2} - \left(\frac{2}{3}\right)^2 = \frac{1}{18} \end{aligned} $$

Next, we introduce some commonly seen continuous probability distributions.