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>

```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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