Question

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

Answer 1

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(x) " if " a < x < b`
`f_2(x) " if " b < x < c`
`0 " elsewhere "`

Then cumulative density function (cdf) is defined as

`F(x) =" " 0 " if " x < a`
`F(x)= int_a^ x f(t)" " dt " if " a le x < b`
`F(x) = F(b) + int_b^x f(t) " "dt " if " b le x < c`
`F(x) = 1 " if " x ge c`.

Note that `F(b) = int_a^ b f_1(t)" " dt`
The cdf just explains how the probability is increasing as we move over the increasing values of `x`.