Density and some other function to find

[Anna Kaladze]

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.

[Stephen Tashi]

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 S(t) satisfying S(0) = 1 ; S(1)= 0 and \frac{dS}{dt} < 0 for 0 < t < 1

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


You didn't give a specific S so I assume an answer must be expressed in terms of S . You used the constant t_0 in your statement of the problem and you wrote S as S(t_0, t) suggesting that S is a function of an interval. This might give some reader a hint about the solution, but I omitted t_0 it since it isn't necessary.