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Re: NIntegrate[UnitStep[...]PDF[...],{x,...}] hard to integrate

• To: mathgroup at smc.vnet.net
• Subject: [mg93468] Re: NIntegrate[UnitStep[...]PDF[...],{x,...}] hard to integrate
• From: Bill Rowe <readnews at sbcglobal.net>
• Date: Mon, 10 Nov 2008 03:30:01 -0500 (EST)

On 11/9/08 at 5:26 AM, erwann.rogard at gmail.com (er) wrote:

>Much helpful, thanks!

>OK, but you have to assume that there are exactly 2 roots to f[x]-
>t,right?

>That may be the case for particular values of t, but for others I
>may have, for example, a left interval (only 1 root) etc. So, what
>if I need  say g[t]==0.95, instead of 0.25?

If there is only one root, then it follows your integral is either

CDF[NormalDistribtuon[0,1/2],x]

or

1 - CDF[NormalDistribution[0,1/2],x]

depending on where you set t. In any case, it will always be
faster to find the appropriate roots of t-f[x] then use the CDF
function to compute g rather than doing the numerical integration.

>and what if I need to carry out these computations for various definitions=
 of f?

That simply changes the part where you find the roots.

When you use UnitStep[t - f[x]] PDF[NormalDistribution

the effect of the UnitStep is to programmatically set the limits
for the integration. Selecting a different f[x] simply selects
different limits, different points where the cumulative
distribution function needs to be evaluated. It is far less
computational effort to find the points where UnitStep[t - f[x]]
is one than it is to do the numerical integration problem.