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MathGroup Archive 2004

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Re: Another Integrate error

  • To: mathgroup at smc.vnet.net
  • Subject: [mg52604] Re: Another Integrate error
  • From: David Bailey <dave at Remove_Thisdbailey.co.uk>
  • Date: Fri, 3 Dec 2004 03:54:51 -0500 (EST)
  • References: <comgts$9gn$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Ray Koopman wrote:
> 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)
> 
Hi,

This looks like the same bug that Vivien Lecomte reported a short while 
ago. It has gone at 5.1.

I guess you have to accept that all software has bugs - including the 
stuff they use to manage air traffic control - and in my experience 
Mathematica has less bugs than most. If a result is important to you - 
or seems unlikely - it is always worth cheching it another way - such as 
  doing the double integral in two steps, or using NIntegrate for one or 
two randomly chosen values of r.

David Bailey


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