Suppose I solve the logistic equation (with $ y(0) = 1$ at $ t=10$ ),
$ $ y(10) = \frac{\kappa e^{10 r}}{\kappa +e^{10 r}-1},$ $
where $ \kappa$ is the carrying capacity and $ r$ is the initial growth rate.
Further suppose that I have the following uncertainty about the inputs: $ r\sim U(0,1)$ and $ \kappa \sim U(10,20)$ . I can simulate what the distribution of $ y(10)$ looks like,
fSampleRK[] := {RandomReal[{0, 1}], RandomReal[{10, 20}]} fLogistic[r_, \[Kappa]_, t_, a_] := (a E^(r t) \[Kappa])/(-a + a E^(r t) + \[Kappa]) fSampleQ[t_, a_] := Block[{lRK = fSampleRK[]}, fLogistic[lRK[[1]], lRK[[2]], t, a]] Histogram[Table[fSampleQ[10, 1], {100000}], 100]
Yielding the monstrosity below,
I rather optimistically tried the following,
aDist = TransformedDistribution[(E^(10 r) \[Kappa])/(-1 + E^(10 r) + \[Kappa]), {r \[Distributed] UniformDistribution[{0, 1}], \[Kappa] \[Distributed] UniformDistribution[{0, 10}]}] PDF[aDist, Q]
But Mathematica falls over and returns the bottom line unevaluated.
Can anyone think of a way to determine the analytic PDF here?
(Of course I can sample and fit a kernel density estimator to the samples, but I am looking for an analytic result.)