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.