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.)