Re: MultinormalDistribution Question
- To: mathgroup at smc.vnet.net
- Subject: [mg120179] Re: MultinormalDistribution Question
- From: Ray Koopman <koopman at sfu.ca>
- Date: Mon, 11 Jul 2011 06:58:57 -0400 (EDT)
- References: <ivbpsr$o5b$1@smc.vnet.net>
On Jul 10, 2:03 am, Steve <s... at epix.net> wrote:
> Hello,
>
> Can someone help me with this ?
>
> I have 2 normal distributions; dist1 describes x and dist2 describes
> y. Each are fully defined and are correlated to one another by the
> correlation coefficient. How can I detemine the mean and standard
> deviation of the expected normal distribution that is associated with
> a given x value from dist1 ?
>
> An example:
> mean1 = 5.8
> sigma1 =0 .2
>
> mean2 = 5.3
> sigma2 = 0.2
>
> Correlation Coefficient, rho = 0.6
>
> Given an x value of 6.3 (from dist1) what is the corresponding mean
> and standard deviation of y ?
>
> I can view the combined density function from the following:
>
> Mu = {mean1, mean2}
> CapSigma = {{sigma1^2, rho*sigma1*sigma2} , {rho, rho*sigma1*sigma2}
> dist = MultinormalDistribution[Mu,CapSigma]
> pdf = PDF[dist,{x,y}]
> plot1 = Plot3D[pdf, {x,4,7},{y,4,7}, PlotRange->All]
>
> but can't see how to determine the mean and the standard deviation of
> y for a given value of x, like 6.3
>
> Any help would be appreciated.
>
> Thanks,
>
> --Steve
The conditional mean of y given x is (x - mx)*r*sy/sx + my,
and the conditional standard deviation is sy*Sqrt[1 - r^2],
where mx & my are the marginal means,
sx & sy are the marginal standard deviations,
and r is the correlation.