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## 'PCDF' printed from http://nrich.maths.org/

Construct a cumulative distribution function $F(x)$ of a random
variable which matches the probability density function of another
random variable whenever $F(x)\neq 1$. How many different sorts can
you make?

Could you make a cdf $G(x)$ which could be used as a pdf for all
values of $x< \infty$ ? Give as clear a reason as
possible.

Can you create an example in which the cumulative distribution
function $F(x)$ of a random variable $X$ and the probability
density function $f(x)$ of the same random variable $X$
are identical whenever $F(x)< 1$?