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

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

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

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?

Re: Non Linear ODE whose solution is can be viewed as a cumulative distribution funct

Quote:

Originally Posted by

**JeffN12345** 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

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

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?

Your ODE is of variables seperable type, so putting $\displaystyle y(x)=F(x) $:

$\displaystyle \int \frac{1}{y^a(1-y)^b}\; dy = \int s\;dx$

with the condition $\displaystyle y(m)=1/2$ determining the arbitary constant.

CB