Let X be continuous a random variable who's support is the entire real line and who's cumulative distribution function satisfies the initial value problem
;
note that a>0, b>0, s>0 and m is real. m is the median of the distribution,
Is it possible to explicitly solve for the CDF, F(x), the PDF f(x)=F'(x), the moment or probability generating functions for X, and/or the inverse function of the CDF?