I think it is meaningful to ask for a non-trivial Pr (X < x, Y=y) when you are writing down the likelihood for parameter estimation. This is commonly the case in likelihood estimation in bivariate failure time models. If one interprets Pr(Y=y) as the density evaluated y then:
Pr(X<x,Y=y) = Pr(X<x | Y=y) * f(y) In R: Pr(X<x,Y=y) = pnorm(x, mu=mu[1] + Sigma[1,2]*(y-mu[2])/Sigma[2,2], sd=sqrt(Sigma[1,1] - (Sigma[1,2]^2)/Sigma[2,2])) * dnorm(y, mu=mu[2], sd=sqrt(sigma[2,2])) Ravi. ---------------------------------------------------------------------------- ------- Ravi Varadhan, Ph.D. Assistant Professor, The Center on Aging and Health Division of Geriatric Medicine and Gerontology Johns Hopkins University Ph: (410) 502-2619 Fax: (410) 614-9625 Email: [EMAIL PROTECTED] Webpage: http://www.jhsph.edu/agingandhealth/People/Faculty/Varadhan.html ---------------------------------------------------------------------------- -------- -----Original Message----- From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On Behalf Of Rolf Turner Sent: Wednesday, October 01, 2008 3:20 PM To: Sasha Pustota Cc: [EMAIL PROTECTED] Subject: Re: [R] Bivariate normal On 2/10/2008, at 4:43 AM, Sasha Pustota wrote: > Package mvtnorm provides dmvnorm, pmvnorm that can be used to compute > Pr(X=x,Y=y) and Pr(X<x,Y<y) for a bivariate normal. > > Are there functions that would compute Pr(X<x,Y=y)? Yes: foo <- function(x,y) { 0 } > I'm currently using "integrate" with dmvnorm but it is too slow. Words fail me ..... see fortune("brain surgery"). I presume you really want Pr(X < x | Y = y) rather than the probability that X is less than x *and* Y equals y. To find this, see any decent textbook on multivariate statistics. (E.g. Morrison.) You can explicitly write down the distribution of X given that Y = y. If (X,Y) is bivariate Gaussian with mean mu and covariance matrix Sigma then *given that* Y = y, X has a Gaussian distribution with mean mu[1] + Sigma[1,2]*(y-mu[2])/Sigma[2,2] and variance equal to Sigma[1,1] - (Sigma[1,2]^2)/Sigma[2,2] Knowing that, you can use pnorm to calculate Pr(X<x | Y=y). cheers, Rolf Turner ###################################################################### Attention:\ This e-mail message is privileged and confid...{{dropped:9}} ______________________________________________ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. ______________________________________________ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.