4.1.1 Probability Distributions for Continuous Variables

Example discretizing depth of lake

probability distribution

Let $X$ be a continuous rv. Then a probability distribution or probability density function (pdf) of $X$ is a function $f\left(x\right)$ such that for any two numbers $a$ and $b$ with $a\le b$ ,

$$P\left(a\le X\le b\right)={\int }_a^{b}f\left(x\right)dx$$

That is, the probability that $X$ takes on a value in the interval $\left[a,b\right]$ is the area above this interval and under the graph of the density function, as illustrated in Figure 4.2. The graph of $f\left(x\right)$ is often referred to as the density curve.

Figure 4.2 $P\left(a\le X\le b\right)=$ the area under the density curve between $a$ and $b$

For $f\left(x\right)$ to be a legitimate pdf, it must satisfy the following two conditions:

1. $f\left(x\right)\ge 0$ for all $x$
2. \begin{align} &{\int}_{-\infty }^{\infty }f\left( x\right) {dx} \\ &= \text{area under the entire graph of } f\left( x\right) \\ &= 1 \end{align}

EX 4.4 probability of angle

Because whenever $0\le a\le b\le 360$ in EX 4.4 probability of angle, $P\left(a\le X\le b\right)$ depends only on the width $b-a$ of the interval, $X$ is said to have a uniform distribution.

uniform distribution

A continuous rv $X$ is said to have a uniform distribution on the interval $\left[A,B\right]$ if the pdf of $X$ is

$$f\left(x;A,B\right)=\{\begin{array}{cc}\frac{1}{B-A}& A\le x\le B\\ 0& \text{ otherwise }\end{array}$$

The graph of any uniform pdf looks like the graph in Figure 4.3 except that the interval of positive density is $\left[A,B\right]$ rather than $\left[0,360\right]$ .

• In the discrete case, a probability mass function (pmf) tells us

how little “blobs” of probability mass of various magnitudes are distributed along the measurement axis.

• In the continuous case, probability density is “smeared” in a continuous fashion along the interval of possible values. When density is smeared uniformly over the interval, a uniform pdf, as in Figure 4.3, results.

Probability of single point:

• When $X$ is a discrete random variable, each possible value is assigned positive probability.
• This is not true of a continuous random variable (that is, the second condition of the definition is satisfied) because the area under a density curve that lies above any single value is zero:

$$\begin{array}{rl}P(X=c)& ={\int }_c^{c}f(x)dx\\ & =\underset{\epsilon \to 0}{\lim }{\int }_{c-\epsilon }^{c+\epsilon }f(x)dx\\ & =0\end{array}$$

The fact that $P\left(X=c\right)=0$ when $X$ is continuous has an important practical consequence:

The probability that $X$ lies in some interval between $a$ and $b$ does not depend on whether the lower limit $a$ or the upper limit $b$ is included in the probability calculation:

$$\begin{array}{rl}P\left(a\le X\le b\right)& =P\left(a<X<b\right)\\ & =P\left(a<X\le b\right)\\ & =P\left(a\le X<b\right)\end{array}\text{\hspace{1em}(4.1)}$$

If $X$ is discrete and both $a$ and $b$ are possible values (e.g., $X$ is binomial with $n=20$ and $a=5,b=10$ ), then all four of the probabilities in (4.1) are different.

physical analog

The zero probability condition has a physical analog. Consider a solid circular rod with cross-sectional area $=1{\mathrm{in}}^2$ . Place the rod alongside a measurement axis and suppose that the density of the rod at any point $x$ is given by the value $f\left(x\right)$ of a density function. Then

• if the rod is sliced at points $a$ and $b$ and this segment is removed, the amount of mass removed is ${\int }_a^{b}f\left(x\right)dx$;
• if the rod is sliced just at the point $c$ , no mass is removed. Mass is assigned to interval segments of the rod but not to individual points.

EX 4.5 time headway

Unlike discrete distributions such as the binomial, hypergeometric, and negative binomial, the distribution of any given continuous rv cannot usually be derived using simple probabilistic arguments. Instead, one must make a judicious choice of pdf based on prior knowledge and available data. Fortunately, there are some general families of pdf’s that have been found to be sensible candidates in a wide variety of experimental situations; several of these are discussed later in the chapter.

Just as in the discrete case, it is often helpful to think of the population of interest as consisting of $X$ values rather than individuals or objects. The pdf is then a model for the distribution of values in this numerical population, and from this model various population characteristics (such as the mean) can be calculated.