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]]/2 is the standard normal cdf.