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Integrating Probability Surface_HOW?

  • To: mathgroup at
  • Subject: [mg22176] Integrating Probability Surface_HOW?
  • From: beintinc at (BEIntInc)
  • Date: Thu, 17 Feb 2000 01:24:08 -0500 (EST)
  • Sender: owner-wri-mathgroup at

I need to find the cumulative probability density contained within an
elliptical region on the bivariate Gaussian surface.  Everything outside the
elliptical region is zero, the interior of the elliptical region is magnitude
1; i.e., the ellipse "windows" the PDF.

To see this, run the script below (Mathematica 4.0) , and you will see the
rotated 4-sigma contour of the PDF, along with an ellipse (dashed).  I used the
additional PDF (ndist1) simply to form the shape of the windowing ellipse.

My problem is, how to find the volume under the dashed ellipse; in other words,
the volume contained within the ellipse is the desired CDF of the rotated PDF. 
(I am probably not precise here, but hopefully it makes sense).

I was hoping I could do

ff = EllipsoidQuantile[ndist1, .3935]
RegionProbability[ndist, ff]

but that does not work.

Any help greatly appreciated,


John Brooks

<< Statistics`MultinormalDistribution`

r = {{0.14606165662744, -0.08743998612085}, {-0.08743998612085, 
ndist = MultinormalDistribution[{0, 0}, r]

pdf = PDF[ndist, {x1, x2}]

BW1 = .00912442396313364^2

BW2 = .01446829793425295^2

r1 = {{BW2, 0}, {0, BW1}};
ndist1 = MultinormalDistribution[{0, 0}, r1]

pdf1 = PDF[ndist1, {x1, x2}]

Show[Graphics[{EllipsoidQuantile[ndist, .9996], Dashing[{.04, .03}], 
      EllipsoidQuantile[ndist1, .3935]}, Axes -> True, GridLines -> Automatic,
     AspectRatio -> Automatic, Frame -> True]]


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