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

  1. #1
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    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?
    Last edited by JeffN12345; January 9th 2012 at 09:17 PM.
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  2. #2
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    Re: Non Linear ODE whose solution is can be viewed as a cumulative distribution funct

    Quote Originally Posted by JeffN12345 View Post
    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
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