       Re: Integrate the Multivariate normal distribution

• To: mathgroup at smc.vnet.net
• Subject: [mg67472] Re: Integrate the Multivariate normal distribution
• From: "Ray Koopman" <koopman at sfu.ca>
• Date: Tue, 27 Jun 2006 03:14:44 -0400 (EDT)
• References: <31541561.1151237675216.JavaMail.root@eastrmwml07.mgt.cox.net> <e7npcb\$bps\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Miguel Lejeune wrote:
> 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?

In a bivariate normal distribution, both marginal distributions are
normal, and the conditional distribution of either variable given the
other is also normal. When the mean vector = {0,0} and the covariance
matrix = {{1,r},{r,1}}, the bivariate pdf can be written as

f[x,y,r] = f[x] * f[(y-r*x)/Sqrt[1-r^2]]/Sqrt[1-r^2],

where f[z] = Exp[-z^2/2]/Sqrt[2Pi] is the standard normal pdf. Then

Integrate[f[x,y,r],{y,-Infinity,u}] = f[z] * F[(u-r*x)/Sqrt[1-r^2]],

where F[z] = Erfc[-z/Sqrt]/2 is the standard normal cdf.

```

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