Vitalis on July 4, 2008 - 3:36am
Yes, if L is a sum of independent exponentials with means Lo and L1, then L has density
(*) 1/(L1-L0)*( exp(-x/L1)-exp(-x/L0) ), for x >= 0.
(Of course, when L0->L1 this will collapse to a Gamma distribution with 2 degrees of freedom).
In the formulas above, I assumed that S is also Gamma(2,s) distributed, which perhaps is harder to give a physical interpretation than for L. If we assume S ~ Exp(lambda) and L density (*), then
E[D(S,L)] = lambda*(1 - 1/(L1-Lo)*(1/(1/lambda + 1/L1) - 1/(1/lambda + 1/Lo)) ).
But perhaps all this fiddling around with different distibutions is of moderate interest: it is probably most important to recognice that the most important thing is to have some kind of uncertainty ("dispersion") in lambda and L. Whether this is the case, I would say, comes down to how sensitive the "output" is to the various assumptions on the distributions, the "output" being whatever the model help us to say about the real world. (In this case, perhaps the expected volume of discoveries in the next few years.)
Another thing we could try to adress is the sample variability: we have focused on the mean of D(S,L), but it should be straight forward to compute condfidence intervals for future discoveries, given assumptions on the distributions of S and L. (Maybe you already did this).
--
Another (minor) comment: i don't think the single dispersive model corresponds to a uniformly distributed L --- in that model, L = Lo deterministically (i.e. point mass 1 at Lo in probability lingo).
In fact, L uniformly distributed on [0,2*L0] (to make E[L] = L0) gives
E[D(S,L)] = labmda*( 1 - 1/(2*Lo) + 1/(2*Lo)*exp(-2*Lo/lambda) ).
I agree with Vitalis about the statement "it is probably most important to recognice that the most important thing is to have some kind of uncertainty ("dispersion") in lambda and L."
The following figure shows how incremental dicoveries basically trend to the same asymptote, independent of the volume distribution.
I would also note that, contrary to Vitalis' suggestion, the Delta "thin seam" volume situated at L0 would have a cumulative discovery profile that looks like this:
DeltaDiscovery(x) = L0*exp(-L0/x)