# 14.2 - Cumulative Distribution Functions

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

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

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

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. $$f(x)$$:

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:

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