MarcumQ and Speeding-up Computations

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• Subject: [mg131379] MarcumQ and Speeding-up Computations
• From: bruce.colletti at gmail.com
• Date: Tue, 16 Jul 2013 05:58:06 -0400 (EDT)
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```Re 9.0.1 under Windows 7.

Here I find the expected value of a MarcumQ function over a 3D-domain whose joint probability density function is f.

How can I more quickly compute the answer 0.930445?

I think the domain causes the long computation time (if I had only one computation, this would be okay--but I have many).  I had hoped simulation would give a good estimate but it didn't.  I can't figure out how to use Compile with Expectation.

My hunch says that any increase in speed will come from using some bizarre fact that applies the MarcumQ function to chi-square tests.  I've looked for such results but have come up dry.

Any ideas to speed up Mathematica computations?  Thanks.

Bruce

Remove@s;
f = ProbabilityDistribution[Piecewise@{{6, x1 + x2 + x3 <= 1}},
{x1, 0, 1}, {x2, 0, 1}, {x3, 0, 1}];

q = {0.1, 0.2, 0.3, 0.4};
t = Array[s, 3];
t = Append[t, 1 - Total@t];

Timing@N@Expectation[
MarcumQ[3/2, Sqrt[ 50 Total@(((q - t)^2)/q)], Sqrt[7.8]],
Most@t \[Distributed] f]

{555.613162, 0.930445}

```

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