       Re: Integrate the Multivariate normal distribution

• To: mathgroup at smc.vnet.net
• Subject: [mg67456] Re: [mg67451] Integrate the Multivariate normal distribution
• From: Bob Hanlon <hanlonr at cox.net>
• Date: Mon, 26 Jun 2006 00:12:57 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```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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