# 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.

\(F(x)=\int_{-\infty}^x f(t)dt\) for −∞ < |

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

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

*f*(*x*) = 3*x*^{2}

for 0 < *x* < 1. What is the cumulative distribution function *F*(*x*)?

### Example

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

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

*f*(*x*) = *x* + 1

for −1 < *x* < 0, and

*f*(*x*) = 1 − *x*

for 0 ≤ *x* < 1. 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* ≤ −1, when −1 < *x* ≤ 0, for 0 < *x* < 1, and for *x* ≥ 1. The definition of *F*(*x*) for *x* ≤ −1 is easy. Since no probability accumulates over that interval, *F*(*x*) = 0 for *x* ≤ −1. Similarly, the definition of *F*(*x*) for *x* ≥ 1 is easy. Since all of the probability has been accumulated for *x* beyond 1, *F*(*x*) = 1 for *x* ≥ 1. Now for the other two intervals:

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

The cumulative distribution function is therefore a concave up parabola over the interval −1 < *x* ≤ 0 and a concave down parabola over the interval 0 < *x* < 1. Therefore, the graph of the cumulative distribution function looks something like this: