Life-span modeling by finite b-lognormals

Abstract

It is well known that the probability density function(pdf) $f_{aX+b} (x)$of the random variable ${aX+b}$where a and b are arbitrary real constants with respect to the independent variable x, is related to the pdf $f_{\rm{X}}(x)$of the random variable X by the linear transformation formula for random variables that reads:

$$ f_{aX+b}(x)=\frac{1}{\mid a \mid} f_X \left(\frac{x-b}{\mid a \mid}\right).$$

(5.1)