Integral of PDF
To really understaand it you will have to look at the fortran code underlying integrate. I tracked it back through a couple of layers (dqagi, dqagie, ... just use google, these are old netlib subroutines) then decided I ought to get back to grading papers :-) It looks like the integral is split into the sum of two integrals, (-Inf,0] and [0,Inf).
integrate(function(x) dnorm(x, 350,50), 0, Inf)
1 with absolute error < 1.5e-05
integrate(function(x) dnorm(x, 400,50), 0, Inf)
1.068444e-05 with absolute error < 2.1e-05
integrate(function(x) dnorm(x, 500,50), 0, Inf)
8.410947e-11 with absolute error < 1.6e-10
integrate(function(x) dnorm(x, 500,50), 0, 1000)
1 with absolute error < 7.4e-05
The integral from 0 to Inf is the lim_{x -> Inf} of the integral
over [0,x]. I suspect that the algorithm is picking an interval
[0,x], evaluating the integral over that interval, and then performing
some test to decide whether to expand the interval. When the initial
interval doesn't contain much, expanding a little won't add enough to
trigger another iteration.
albyn
On Thu, Dec 02, 2010 at 04:21:34PM -0500, Doran, Harold wrote:
The integral of any probability density from -Inf to Inf should equal 1, correct? I don't understand last result below.
integrate(function(x) dnorm(x, 0,1), -Inf, Inf)
1 with absolute error < 9.4e-05
integrate(function(x) dnorm(x, 100,10), -Inf, Inf)
1 with absolute error < 0.00012
integrate(function(x) dnorm(x, 500,50), -Inf, Inf)
8.410947e-11 with absolute error < 1.6e-10
all.equal(integrate(function(x) dnorm(x, 500,50), -Inf, Inf)$value, 0)
[1] TRUE
sessionInfo()
R version 2.10.1 (2009-12-14) i386-pc-mingw32 locale: [1] LC_COLLATE=English_United States.1252 LC_CTYPE=English_United States.1252 [3] LC_MONETARY=English_United States.1252 LC_NUMERIC=C [5] LC_TIME=English_United States.1252 attached base packages: [1] stats graphics grDevices utils datasets methods base other attached packages: [1] statmod_1.4.6 mlmRev_0.99875-1 lme4_0.999375-35 Matrix_0.999375-33 lattice_0.17-26 loaded via a namespace (and not attached): [1] grid_2.10.1 nlme_3.1-96 stats4_2.10.1 tools_2.10.1 [[alternative HTML version deleted]]
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Albyn Jones Reed College jones at reed.edu