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Re: How to express the results in normal cdf instead of erf()

  • To: mathgroup at smc.vnet.net
  • Subject: [mg97133] Re: How to express the results in normal cdf instead of erf()
  • From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
  • Date: Fri, 6 Mar 2009 04:24:09 -0500 (EST)
  • Organization: Uni Leipzig
  • References: <goo7jt$sga$1@smc.vnet.net>
  • Reply-to: kuska at informatik.uni-leipzig.de

Hi,

res = Integrate[PDF[NormalDistribution[0, 1], x] Exp[x], x];

and

res /.  Erf[a_] :> Erf[Expand[a]] /.
  RuleDelayed @@
   Reverse[((2* # - 1) & /@ {cdf[NormalDistribution[0, 1], a*x + b],
         1/2*(1 + Erf[a_.*x/Sqrt[2] + b_.])} /. x -> Sqrt[2]*y) /.
     y -> x]

you must use cdf[] instead of CDF[] because otherwise Mathematica will
evaluate it back to Erf[]

Regards
   Jens

Irving Zhu wrote:
> Hi All,
> 
> I am working on some integrals involving the Normal Distribution, and would
> like the end results to be expressed in normal density as opposed to the
> Erf[] function.
> 
> For example, the default result of the following:
> 
> Integrate[PDF[NormalDistribution[0, 1], x] Exp[x], x] will result in an
> expression in Erf[].
> 
> I'd like it to be in terms of CDF[NormalDistribution[]] instead.
> 
> Of course the two functions are related:
> 
> CDF[NormalDistribution[0,1],x] ==1/2*(1+Erf[[x/Sqrt[2]])
> 
> 
> Is there any way to do that?
> 
> Thanks,
> Irving.
> 
> 


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