@tomleslie Thanks for shedding valuable light on the situation, Tom.

@Carl Love The hypergeom appears to be fast and robust, with the `p` value converted to exact rational before passing to NextZero.

Here is a procedure for it, and some timings, and a loglog plot as Carl showed.
 

Download Quant01.mw

@mmcdara I don't understand your point. You used 2 instead of 1.965.

Using 2 instead of 1.965 then the regular Statistics package find the same results quickly. And using 1.965 instead of 2 and the Student:-Statistics package also take a long time...

restart:
with(Statistics):
X := RandomVariable(NegativeBinomial(2, 0.003)):

Quantile(X,0.98, numeric);
                             1941.

X := RandomVariable(NegativeBinomial(2, 3e-10)):
Quantile(X,0.98, numeric);
                                        10
                     1.94464056686655 10  

restart:
with(Student:-Statistics):
X := NegativeBinomialRandomVariable(1.965, 0.003):

Quantile(X,0.98, numeric); # goes away...

@David1 The GlobalSearch command searches for local and global extrema, and hopefully finds global extrema. If you pass it the option solutions=1 then it may exit as soon as it finds one. I can imagine that that would decrease the chance of its finding a global extremum. A reasonable interpretation of the help page for this command is that passing solutions=1 could allow it to return as soon as it found a single local extreme point, which is certainly not at all your goal.

If you want DirectSearch to try and find a single point which is the global extremum then why not use its command GlobalOptima?  That command's goal is to search for the best objective value. As you will see on its help page that command takes many options, because (as has been pointed out to you before) there is no foolproof method for computing the globally best result of a general, nonconvex, nonlinear (and possibly discontinuous) optimization problem.

If you want to increase the chances that it finds the global extremum then make it work harder, searching from more randomized starting points and with a longer time-limit and evaluation-limit. That relates to its options number, timelimit, and evaluationlimit. But other options can also come into play. Read the help page carefully. Experiment with the options.

@jamalator How can y and q both be prime and satisfy 8695*y = 1341*q ?

@Joe Riel Since this discussion relates to thismodule then perhaps it's on-topic to mention function:-somestaticexport. (or the subtle issues in invoking :-eval(self) or mapping a commonly named static across different objects in a list -- including doing so in tricky situations).

Sorry, Carl. That was me. Unfortunately there's no way at present for me to toggle the Product entry to a specific Maple version without Mapleprimes considering the Question as being edited. I figured that if I did them in a block it might be more clear that it was a series of no-ops. I try not to do that sort of edit often.

@Melvin Brown You're welcome, Marvin.

@waseem What do you want to combine? With respect to what do you want to collect? Why are you not being clear?

Perhaps you are looking for something like this form?

Download waseem_2.mw

If this was your actual goal then why didn't you say so in your original question?

@Carl Love I didn't have a particularly strong reason, except as a single sanity check. (I'm sometimes a little hesitant for trig integrals over a symmetric range about zero, slightly more so when I see the multiple of Pi on the argument.)

I see that Rouben found the same thing, while I was working on it and submitting.

Yet another (much slower) approach that works here is to change variables to make the inner integral range from a=1..infinity, and then do purely numeric integration. Mariusz's nice workaround also serves as another corroboration.

restart;
with(IntegrationTools):
H:=Int(exp(-a^2-b^2-c^2),
       [a=b^2/(4*c)..infinity, b=-infinity..infinity, c=0..infinity]):
T:=ExpandMultiple(H):
T1:=Change(op([1,1],T),a=s*(b^2/(4*c)),[s])
    assuming c>0, b::real:
new:=CollapseNested(Int(Int(T1,op([1,2],T)),op(2..,T))):
evalf(op(0,new)(op(new),epsilon=1e-5));
                          0.9786008269

@waseem The purpose of procedure H is to temporarily freeze the factors that are nontrivially polynomial in r, so that they don't get expanded.

A simpler procedure than I had above is just this:

H:=proc(ee)
  subsindets(ee,And(polynom(numeric,r),`+`),freeze);
end proc:

somesimplification2.mw

@gal Please upload a short Worksheet or Document that demonstrates the problem and attach it to a Reply/Comment.

You can use the green up-arrow in the Mapleprimes editor to upload an attachment.

@Ronan In the unix world it is an old convention to give subdirectories 3-letter names like bin, doc, lib, src, tmp, etc.

The name "src" stood for source.

But it's rather arbitrary.

My point was that perhaps it's not best to store your .mpl plaintext alongside any .mla archives, if only because it (slightly) confuses the special relationship the directory might have to libname and the .mla files. Not a huge deal.