14.2  Cumulative Distribution Functions
14.2  Cumulative Distribution FunctionsYou 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 nondecreasing step function. For continuous random variables, \(F(x)\) is a nondecreasing continuous function.
Example 142 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 143 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 144
Suppose the p.d.f. of a continuous random variable \(X\) is defined as:
\(f(x)=\begin{cases} x+1, & 1<x<0\\ 1x, & 0\le x<1 \end{cases} \)
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:
\(\begin{equation}F(x)=\left\{\begin{array}{ll}
0, & \text { for } x \leq1 \\
\frac{1}{2}(x+1)^{2}, & \text { for }1<x \leq 0 \\
1\frac{(1x)^{2}}{2}, & \text { for } 0<x<1 \\
1, & \text { for } x \geqslant 1
\end{array}\right.\end{equation}\)
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: