Cumulative Distribution Functions

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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.

Definition. 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 −∞ < x < ∞.

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) = 3x2

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):

Picture of 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:

Cumulative distribution function

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:

Graph of CDF