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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!

    Anna.
     

    Attached Files:

  2. jcsd
  3. Sep 23, 2011 #2

    Stephen Tashi

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    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
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