Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Density and some other function to find

  1. Sep 12, 2011 #1
    Hi All,

    I have been trying to unsuccessfully crack a certain problem for my research, but I get stuck. I found it is easy to describe the problem in a separate document. Can you please have a look at the attached file and give me some help? Thanks!


    Attached Files:

  2. jcsd
  3. Sep 23, 2011 #2

    Stephen Tashi

    User Avatar
    Science Advisor

    You'd probably get more responses if you posted in LaTex and stated the question concisely. As I interpret the question, a change of variables would reduce it to the following:

    Given the following:
    The function [itex] S(t) [/itex] satisfying [itex] S(0) = 1 ; S(1)= 0 [/itex] and [itex] \frac{dS}{dt} < 0 [/itex] for [itex] 0 < t < 1 [/itex]

    Find the following:
    Constants [itex] X_{min} < X_{max} [/itex]
    A random variable [itex] X [/itex] that has support [itex] [X_{min}, X_{max}] [/itex].
    and a function [itex] F(S,X) [/itex]
    that will satisfy these conditions:
    [itex] F(S(0),x) = 1 [/itex] for all [itex] X_{min} \leq x \leq X_{max} [/itex]
    [itex] F(S(1),x) = 0 [/itex] for all [itex] X_{min} \leq x \leq X_{max} [/itex]
    [itex] \frac{\partial F(S,x)}{\partial x } < 0 [/itex] for [itex] 0 < t < 1 [/itex]
    For each [itex] 0 < t < 1 [/itex], [itex] \int_{X_{min}}^{X_{max}} F(S(t),X) dX = S(t) [/itex]

    You didn't give a specific [itex] S [/itex] so I assume an answer must be expressed in terms of [itex] S [/itex]. You used the constant [itex] t_0 [/itex] in your statement of the problem and you wrote [itex] S [/itex] as [itex] S(t_0, t) [/itex] suggesting that S is a function of an interval. This might give some reader a hint about the solution, but I omitted [itex] t_0 [/itex] it since it isn't necessary.
    Last edited: Sep 23, 2011
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook