# [R] Bivariate normal

Sasha Pustota popgen at gmail.com
Wed Oct 1 20:04:47 CEST 2008

Thanks Jay. I realized that I was doing it a silly way shortly after I
posted and that the answer i was looking for is simply

condXY(y, x, my, mx, r) * dnorm(y, my)

condXY <- function(y, x, my, mx, r) {
m <- mx + r*(y - my)
s <- sqrt(1-r^2)
p <- 1 - pnorm(x, mean=m, sd=s) + pnorm(-x, mean=m, sd=s)
}

On Wed, Oct 1, 2008 at 1:11 PM, G. Jay Kerns <gkerns at ysu.edu> wrote:
> Dear Sasha,
>
> On Wed, Oct 1, 2008 at 11:43 AM, Sasha Pustota <popgen at gmail.com> 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)?
>> I'm currently using "integrate" with dmvnorm but it is too slow.
>
>
> Strictly speaking, the probability that you are asking to calculate is
> always 0, for every value of y.  The reason is that the quantity you
> are requesting is the _volume_ of a vertical slice, at the value y,
> which is zero.  It may be useful to think carefully about the problem
> you are trying to solve...  perhaps a conditional probability is more
> appropriate.
>
> You did not say exactly which integral you are trying to compute:
> conceivably it would be
>
> \int_{-\infty}^{x}  f(u, y) du,
>
> where f(.,.) is the bivariate normal pdf.  If this is indeed what you
> want, then a work-around would be to calculate P( X < x | Y = y ).  We
> know that given Y=y, X is normal with mean and variance formulas given
> in most introductory statistics books.  Thus, you could compute P( X <
> x | Y = y ) with pnorm(x, mean = something, sd = something).
>
> In that case, the integral above would simply be P( X < x | Y=y ) *
> f(y), where f(y) is the marginal pdf of Y (a dnorm).
>
> Note that the above is assuming that y is a fixed constant;  if not,
> then you may want to check out the Ryacas package.
>
> I hope that this helps,
> Jay
>
>
>
>
>
> ***************************************************
> G. Jay Kerns, Ph.D.
> Associate Professor
> Department of Mathematics & Statistics
> Youngstown State University
> Youngstown, OH 44555-0002 USA
> Office: 1035 Cushwa Hall
> Phone: (330) 941-3310 Office (voice mail)
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> E-mail: gkerns at ysu.edu
> http://www.cc.ysu.edu/~gjkerns/
>