#### Cumulative Distribution Function

- In 3.1 Continuous Probability Distribution and Density Function we understand that probability density functions are

f(x)=\operatorname{Pr}(X=x)

and probability mass functions (for discrete random variables) is p(x)=\operatorname{Pr}(X=x). In the continuous case, a more useful way of describing a random variable is the cumulative distribution function (c.d.f.).

- A c.d.f. function is defined as follows:

F(x)=\operatorname{Pr}(X \leq x)=\int_{-\infty}^{x} f(t) d t=\int_{c}^{x} f(t) d t

where f is defined over the interval [c,\ d]. Therefore, f(x) denote the p.d.f. of a random variable, and we denote F(x) as its c.d.f.

Note: The expression \int_{-\infty}^{x} f(t) d t=\int_{c}^{x} f(t) d t indicates that f(x)=0 when x does not belong to [c,\ d]. See the example below for details.