14.2 - Cumulative Distribution Functions

You might recall that the cumulative distribution function is defined for discrete random variables as:

\(F(x)=P(X\leq x)=\sum\limits_{t \leq x} f(t)\)

Again, \(F(x)\) accumulates all of the probability less than or equal to \(x\). The cumulative distribution function for continuous random variables is just a straightforward extension of that of the discrete case. All we need to do is replace the summation with an integral.

Cumulative Distribution Function ("c.d.f.")

The cumulative distribution function ("c.d.f.") of a continuous random variable \(X\)is defined as:

\(F(x)=\int_{-\infty}^x f(t)dt\)

for \(-\infty<x<\infty\).

You might recall, for discrete random variables, that \(F(x)\) is, in general, a non-decreasing step function. For continuous random variables, \(F(x)\) is a non-decreasing continuous function.

Example 14-2 Revisited Section

Let's return to the example in which \(X\) has the following probability density function:

\(f(x)=3x^2, \qquad 0<x<1\)

What is the cumulative distribution function \(F(x)\)?

Example 14-3 Revisited again Section

Let's return to the example in which \(X\) has the following probability density function:

\(f(x)=\dfrac{x^3}{4}\)

for \(0<x<2\). What is the cumulative distribution function of \(X\)?

Example 14-4 Section

Suppose the p.d.f. of a continuous random variable \(X\) is defined as:

\(f(x)=\begin{cases} x+1, & -1<x<0\\ 1-x, & 0\le x<1 \end{cases} \)

Find and graph the c.d.f. \(F(x)\).

Solution

If we look at a graph of the p.d.f. \(f(x)\):

Picture of p.d.f. f(x)

we see that the cumulative distribution function \(F(x)\) must be defined over four intervals — for \(x\le -1\), when \(-1<x\le 0\), for \(0<x<1\), and for \(x\ge 1\). The definition of \(F(x)\) for \(x\le -1\) is easy. Since no probability accumulates over that interval, \(F(x)=0\) for \(x\le -1\). Similarly, the definition of \(F(x)\) for \(x\ge 1\) is easy. Since all of the probability has been accumulated for \(x\) beyond 1, \(F(x)=1\) for \(x\ge 1\). Now for the other two intervals:

In summary, the cumulative distribution function defined over the four intervals is:

\(\begin{equation}F(x)=\left\{\begin{array}{ll}
0, & \text { for } x \leq-1 \\
\frac{1}{2}(x+1)^{2}, & \text { for }-1<x \leq 0 \\
1-\frac{(1-x)^{2}}{2}, & \text { for } 0<x<1 \\
1, & \text { for } x \geqslant 1
\end{array}\right.\end{equation}\)

The cumulative distribution function is therefore a concave up parabola over the interval \(-1<x\le 0\) and a concave down parabola over the interval \(0<x<1\). Therefore, the graph of the cumulative distribution function looks something like this:

Graph of CDF