# 8.7 | Distribution Functions

The function, $$p(x)$$ , is a Probability density function, or pdf, if
$$\int_{-\infty}^\infty p(x) \; dx = 1 \text{ and } p(x) \geq 0 \text{ for all } x.$$
The fraction of the population for which $$x$$ is between $$a$$ and $$b$$ is equal to the area under the graph of $$p$$ between $$a$$ and $$b$$, which is given by
$$\int_a^b p(x) \; dx.$$
A cumulative distribution function, or cdf, $$P(t)$$, of a density function $$p$$ , is defined by
$$P(t) = \int_{-\infty}^t p(x) \; dx$$
and represents the fraction of population having values of $$x$$ below $$t$$. Thus, $$P$$ is an antiderivative of $$p$$. That is, $$P’ = p$$.
Any cumulative distribution has the following properties:

• $$P$$ is increasing (or nondecreasing).
• $$\displaystyle \lim_{t \rightarrow \infty} P(t) = 1$$ and $$\displaystyle \lim_{t \rightarrow -\infty} P(t) = 0$$.
• $$\displaystyle \int_a^b p(x) \; dx = P(b) – P(a)$$.