Integral of PDF

Albyn's reply is in line with an hypothesis I was beginning to
formulate (without looking at the underlying FoRTRAN code),
prompted by the hint in '?integrate':

  Details:
     If one or both limits are infinite, the infinite range
     is mapped onto a finite interval.
     For a finite interval, globally adaptive interval
     subdivision is used in connection with extrapolation
     by the Epsilon algorithm.

as a result of some numerical experiments. Following up on
Harold Doran's original dnorm(x,mean.mean/10) pattern (and
quoting a small subset of the experiments ... ):

  integrate(function(x) dnorm(x, 100,10), -Inf, Inf)
  # 1 with absolute error < 0.00012
  integrate(function(x) dnorm(x, 200,20), -Inf, Inf)
  # 1.429508e-08 with absolute error < 2.8e-08

  integrate(function(x) dnorm(x, 140,14), -Inf, Inf)
  # 1 with absolute error < 2.2e-06
  integrate(function(x) dnorm(x, 150,15), -Inf, Inf)
  # 2.261582e-05 with absolute error < 4.4e-05

  integrate(function(x) dnorm(x, 144,14.4), -Inf, Inf)
  # 1 with absolute error < 1.7e-06
  integrate(function(x) dnorm(x, 145,14.5), -Inf, Inf)
  # 5.447138e-05 with absolute error < 0.00011

  integrate(function(x) dnorm(x, 150,15), -33000, 33300)
  # 1 with absolute error < 0.00012
  integrate(function(x) dnorm(x, 150,15), -34000, 34300)
  # 5.239671e-05 with absolute error < 0.00010

  integrate(function(x) dnorm(x, 150,15),
            -33768.260234277, 34068.2602334277)
  # 0.5000307 with absolute error < 6.1e-05
  integrate(function(x) dnorm(x, 150,15),
            -33768.260234278, 34068.2602334278)
  # 6.139415e-05 with absolute error < 0.00012

So it seems that, depending on how integrate() "maps to a finite
interval", and on how the algorithm goes about its "adaptive
interval subdivision", there are critical points where integrate()
switches from one to another of the following:

[A] The whole of a finite interval containing all but the extreme
      outer tails of dnorm() is integrated over;
[B] The whole of a finite interval containing one half of the
      distribution of dnorm() is integrated over;
[C] The whole of a finite interval lying in the extreme of one
      tail of the dnorm distribution is integrated over.

with result: [A] close to 1; [B] close to 0.5; [C] close to 0.

This is compatible with Albyn's conclusion that the integral is
split into the sum of (-Inf,0) and (0,Inf), with the algorithm
ignoring one, or the other, or both, or neither!

This must be one of the nastiest exemplar's of "rounding error" ever!!!
Ted.

Integral of PDF

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