Non Linear ODE whose solution is can be viewed as a cumulative distribution function

[Jeff.N]

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

F'(x)=s$\cdot$F(x)^a$\cdot$(1-F(x))^b
F(m)=1/2

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

 

[JJacquelin]

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

F'(x)=s$\cdot$F(x)^a$\cdot$(1-F(x))^b
F(m)=1/2

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

It is possible to solve the ODE. The result is x as a function of F thanks to the Incomplete Beta Function. Then F(x) is the inverse function.