[R] Bivariate normal

[Rolf Turner]

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

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