# How do you derive a cumulative density function from a probability density function?

Oct 27, 2015

Finding the cumulative density function is a process of integrating the pdf over different intervals over which the random variable is defined.

#### Explanation:

Consider probability density function (pdf) defined as

${f}_{1} \left(x\right) \text{ if } a < x < b$
${f}_{2} \left(x\right) \text{ if } b < x < c$
$0 \text{ elsewhere }$

Then cumulative density function (cdf) is defined as

$F \left(x\right) = \text{ " 0 " if } x < a$
$F \left(x\right) = {\int}_{a}^{x} f \left(t\right) \text{ " dt " if } a \le x < b$
$F \left(x\right) = F \left(b\right) + {\int}_{b}^{x} f \left(t\right) \text{ "dt " if } b \le x < c$
$F \left(x\right) = 1 \text{ if } x \ge c$.

Note that $F \left(b\right) = {\int}_{a}^{b} {f}_{1} \left(t\right) \text{ } \mathrm{dt}$
The cdf just explains how the probability is increasing as we move over the increasing values of $x$.