12.2: Continuous random variables:
Probability distribution functions

Given a sequence of data points $a_1, \ldots, a_n$, its cumulative distribution function $F(x)$ is defined by

That is, $F(A)$ is the relative proportion of the data points taking value less than or equal to $A$.

Properties of cumulative distribution functions

• The cumulative distribution function $F$ for the data points $a_1, \ldots, a_n$ may be computed from the corresponding random variable $X$ via the formula
  
  $$F(A) = \sum_{v \leq A} v X(v)$$
  
  

• $\lim_{A \to -\infty} F(A) = 0$ and $\lim_{A \to \infty} F(A) = 1$
• $A \leq B \Rightarrow F(A) \leq F(B)$

Example

Given the data points $5, 3, 6, 2, 5, 2, 1, -4, 0, 4, 9, 10, 3, 3, 6, 8$, compute $F(4)$ where $F(x)$ is the corresponding cumulative distribution function.

Solution

There are a total of sixteen data points of which nine have a value less than or equal to four. Thus, $F(4) = \frac{9}{16}$.

Computing probabilities with cumulative distributions

One may regard the cumulative distribution function $F(x)$ as describing the probability that a randomly chosen data point will have value less than or equal to $x$.

If $X$ is the correponding random variable, one often writes

$$\mathrm{Pr}(X \leq x) = F(x)$$

From $F$ we may compute other probabilities. For instance, the probability of obtaining a value greater than $A$ but less than or equal to $B$ is

$$\mathrm{Pr}(A < X \leq B) = F(B) - F(A)$$

Continuous random variables

We may wish to express the probability that a numerical value of a particular experiment lie with a certain range even though infinitely many such values are possible.

• Express the probability that if a coin is flipped repeatedly, the first result of heads will occur by the $n^\text{th}$ flip.
• What is the probability that a major earthquake will occur on the North Hayward fault within the next five years?
• What is the probability that a randomly selected high school senior will score at least $600$ on the SAT?

General cumulative distribution functions

A cumulative distribution function (in general) is a function $F(x)$ defined for all real numbers for which

• $A \leq B \Rightarrow F(A) \leq F(B)$
• $\lim_{x \to -\infty} F(x) = 0$
• $\lim_{x \to \infty} F(x) = 1$

We write $X$ for the corresponding random variable and treat $F$ as expressing $F(A) = \text{ the probability that } X \leq A = \mathrm{Pr}(X \leq A)$.

Probability densities

If the cumulative distribution function $F(x)$ (for the random variable $X$) is differentiable and have derivative $f(x) = F'(x)$, then we say that $f(x)$ is the probability density function for $X$.

For numbers $A \leq B$ we have

$$\begin{eqnarray*} \mathrm{Pr}(A < X \leq B) & = & F(B) - F(A) \\ & = & \int_A^B f(x) dx \end{eqnarray*}$$

Properties of probability densities

• $0 \leq f(x)$ for all values of $x$ since $F$ is non-decreasing.
• $F(A) = \int_{-\infty}^A f(x) dx$
• $\lim_{x \to -\infty} f(x) = 0 = \lim_{x \to \infty} f(x)$
• $\int_{-\infty}^\infty f(x) dx = 1$

Conversely, any function satisfying the above properties is a probability density.

Example

The function

is a probability density (for the random variable $X$).

Compute $\mathrm{Pr}(-10 \leq X \leq 10)$.

Solution

We know

$$\begin{eqnarray*} \mathrm{Pr}(-10 \leq X \leq 10) & = & \int_{-10}^{10} f(x) dx ... ... & = & 0 + (-e^{-x} \vert_{x=0}^{x=10}) \\ & = & 1 - e^{-10} \end{eqnarray*}$$