       Re: Integrate the Multivariate normal distribution

• To: mathgroup at smc.vnet.net
• Subject: [mg67466] Re: [mg67451] Integrate the Multivariate normal distribution
• From: "Miguel Lejeune" <mlejeune at andrew.cmu.edu>
• Date: Mon, 26 Jun 2006 00:54:17 -0400 (EDT)
• References: <31541561.1151237675216.JavaMail.root@eastrmwml07.mgt.cox.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Thank you Bob.

I have one more question, also related to the bivariate normal distribution.

I would like to compute (numerically) the following integrale (PDF is the
probability density function of the bivariate normal distribution):

Integrale of  [PDF[ndist, {x1, x2}] with respect to x2 (dx2) and the
integration bounds are -Infinity and 1.
with
ndist = MultinormalDistribution[{0, 0}, r];
r = {{1, 0.2}, {0.2, 1}};

Could you indicate me how I should do?

Many thanks,
Miguel

----- Original Message -----
From: "Bob Hanlon" <hanlonr at cox.net>
To: mathgroup at smc.vnet.net
Subject: [mg67466] Re: [mg67451] Integrate the Multivariate normal distribution

> You need to load the package.
>
> Off[General::spell1];
>
> Needs["Statistics`MultinormalDistribution`"];
>
> r={{1,0.2},{0.2,1}};
>
> ndist=MultinormalDistribution[{0,0},r];
>
> pdf=PDF[ndist,{x1,x2}]
>
> 0.16243683359034922*
>  E^((1/2)*((-x1)*(1.0416666666666667*x1 -
>       0.20833333333333337*x2) -
>     x2*(1.0416666666666667*x2 - 0.20833333333333337*x1)))
>
> ContourPlot[pdf,{x1,-2,2},{x2,-2,2}];
>
> NIntegrate[CDF[ndist,{x1,x2}],
>  {x1,-Infinity,0},{x2,-Infinity,0}]
>
> 0.212349
>
>
> Bob Hanlon
>
> ---- Miguel Lejeune <mlejeune at andrew.cmu.edu> wrote:
>>   Hello,
>>
>>   I am using the MultiNormal function to compute the probability density
>> function (pdf_ and cumulative probability distribution (cdf) of a
>> bivraite normally distributed variable.
>>
>>   I have two questions.
>>
>>   1) I followed the help file to get familiar with that function.
>>   But although I repeat what is indicated, I do not obtain the same
>> output. Could you please indicate me?
>>   Example:
>>
>>   In:  Statistics`MultinormalDistribution
>>   In: (r = {{1, 0.2}, {0.2, 1}};
>>   ndist = MultinormalDistribution[{0, 0}, r])
>>
>>   I obtain as output:
>>   Out: MultinormalDistribution [ {{0, 0}, {1, 0.2}, {1,0.2, 1}}]
>>   which is fine.
>>
>>   However, when I type:
>>
>>   In:  pdf = PDF[ndist, {x1, x2}]
>>
>>   The only output I obtain is:
>>   MultinormalDistribution[{0, 0}, {{1, 0.2}, {1, 0, 2}}]
>>
>>   while in the help file it is indicated that I should obtain an
>> algebraic expression.
>>
>>   Why is it??
>>
>>
>>   2) My second question. I would like to proceed to the numerical
>> integration of the CDF of the bivariate normal distribution. I enter:
>>
>>   In:   NIntegrate[CDF[ndist, {x1, x2}], {x1, -Infinity, 0},
>> {x2,-Infinity, 0}]
>>
>>   I systematically obtain the following error message:
>>
>>   "NIntegrate::inum: Integrand CDF[MultinormalDistribution[{0, 0},
>> {{1,0.5}, \
>>   {1, 0, 5}}], {x1, x2}] is not numerical at {x1, x2} = {-1., -1.}
>>   "
>>   I do not understand why it is saying that the expression  is not
>> numerical at {-1,1}. Could anybody help?
>>
>>
>>
>>   Many thanks,
>>
>>
>>
>>   Miguel
>>
>>
>
>
>

```

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