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MathGroup Archive 2004

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Re: Another Integrate error

  • To: mathgroup at smc.vnet.net
  • Subject: [mg52589] Re: [mg52573] Another Integrate error
  • From: Bob Hanlon <hanlonr at cox.net>
  • Date: Fri, 3 Dec 2004 03:53:55 -0500 (EST)
  • Reply-to: hanlonr at cox.net
  • Sender: owner-wri-mathgroup at wolfram.com

$Version

5.1 for Mac OS X (October 25, 2004)

Clear[x,y,r];

f[x_,y_,r_]=Exp[(x^2+y^2-2*r*x*y)/(r^2-1)]/(Pi Sqrt[1-r^2]);

Integrate[f[x,y,r],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]

1

Integrate[f[x,y,r],{
  x,-Infinity,Infinity},{y,-Infinity,Infinity},Assumptions->{Element[r,
    Reals],-1<r<1}]

1

Needs["Statistics`MultinormalDistribution`"];

dist=PDF[MultinormalDistribution[{0,0}, {{1,r},{r,1}}],{x,y}]//Simplify

E^((x^2 - 2*r*y*x + y^2)/(2*(r^2 - 1)))/
  (2*Pi*Sqrt[1 - r^2])

Integrate[dist,{x,-Infinity,Infinity},{y,-Infinity,Infinity}]

1


Bob Hanlon

> 
> From: koopman at sfu.ca (Ray Koopman)
To: mathgroup at smc.vnet.net
> Date: 2004/12/02 Thu AM 02:21:45 EST
> To: mathgroup at smc.vnet.net
> Subject: [mg52589] [mg52573] Another Integrate error
> 
> This is a bivariate normal probability density:
> 
> In[1]:= f[x_,y_,r_] = Exp[(x^2 + y^2 - 2*r*x*y)/(r^2 - 1)] / 
>                       (Pi Sqrt[1 - r^2]);
> 
> It should integrate to 1:
> 
> In[2]:= Integrate[f[x,y,r],{x,-Infinity,Infinity},{y,-Infinity,Infinity}]
> Out[2]= 0
> 
> In[3]:= Integrate[f[x,y,r],{x,-Infinity,Infinity},{y,-Infinity,Infinity}, 
>                   Assumptions -> {Element[r,Reals], -1 < r < 1}]
> Out[3]= 1
> 
> In this case I happened to know that the integal should be 1,
> but what about more obscure cases? 
> When can I believe a result from Integrate? 
> 
> In[4]:= $Version
> Out[4]= 5.0 for Mac OS X (November 19, 2003)
> 
> 


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