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

• January 9th 2012, 09:07 PM
JeffN12345
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
$F'(x)=s \cdot F(x)^a\cdot(1-F(x))^b$ ; $F(m)=1/2$

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?
• January 10th 2012, 12:56 AM
CaptainBlack
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
$F'(x)=s \cdot F(x)^a\cdot(1-F(x))^b$ ; $F(m)=1/2$

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?

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

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

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

CB