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 \(\displaystyle \lim_{t \rightarrow -\infty} P(t) = 0\).
  • \(\displaystyle \int_a^b p(x) \; dx = P(b) – P(a)\).

E 8.7 Exercises