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Re: Mathematica Integration help

  • To: mathgroup at smc.vnet.net
  • Subject: [mg86138] Re: Mathematica Integration help
  • From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
  • Date: Mon, 3 Mar 2008 04:40:08 -0500 (EST)
  • References: <fqetdm$jjs$1@smc.vnet.net>

Hi,

look what happens when you compute from:
res = Integrate[
   Product[1/(E^((Subscript[x, i] - \[Mu])^2/(2*\[Sigma]^2))*
       Sqrt[2*Pi*\[Sigma]^2]), {i, 1, n}], \[Mu]]

D[res,\[Mu]]

and see that the result is *not*

Product[1/(E^((Subscript[x, i] - \[Mu])^2/(2*\[Sigma]^2))*
       Sqrt[2*Pi*\[Sigma]^2]), {i, 1, n}]

this obvious a bug.

Regards
   Jens


Francogrex wrote:
> Hi I'm new to Mathematica, I don't know if there are other help forums
> beside this one. I need help estimating the integrated likelihood
> function with mathematica 6.
> 
> My function is below(all equations below are in mathematica raw input
> form, pls convert to standard form to see how they really look like):
> 
> Integrate[Product[1/(E^((Subscript[x, i] - \[Mu])^2/(2*\[Sigma]^2))*
>      Sqrt[2*Pi*\[Sigma]^2]), {i, 1, n}], \[Mu]]
> 
> 
> The result that it's giving me is the following:
> 
> -(2^(-(1/2) - n/2)*
>     E^((n*(-\[Mu] + Subscript[x, i])^2)/(2*\[Sigma]^2))*
>        Pi^(1/2 - n/2)*\[Sigma]*
>        (Sqrt[\[Sigma]^2]/(E^((-\[Mu] +
>                Subscript[x, i])^2/(2*\[Sigma]^2))*\[Sigma]^2))^n*
>        Erf[(Sqrt[n]*(-\[Mu] + Subscript[x, i]))/(Sqrt[2]*\[Sigma])])/
>  Sqrt[n]
> 
> 
> But I know from a publication I've read that the result doesn't
> involve the Erf, but it's the following:
> 
> 1/(E^(Sum[(Subscript[x, i] - OverBar[x])^2, {i, 1,
>         n}]/(2*\[Sigma]^2))*
>       (Sqrt[n]*(2*Pi*\[Sigma]^2)^((n - 1)/2)))
> 
> where:  OverBar[x] = Sum[Subscript[x, i], {i, 1, n}]/n
> 
> Is there any way to get Mathematica to output that? Thanks
> 


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