[R] Bivariate normal

[G. Jay Kerns]

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/