       Re: MultinormalDistribution Question

• To: mathgroup at smc.vnet.net
• Subject: [mg120189] Re: MultinormalDistribution Question
• From: FranD <seacrofter001 at yahoo.com>
• Date: Tue, 12 Jul 2011 06:59:13 -0400 (EDT)

```ClearAll["Global`*"]

dist = MultinormalDistribution[{meanx, meany}, {{sx^2, rho*sx*sy}, {rho*sx*sy, sy^2}}];

pdf = PDF[dist, {x, y}];

i = Integrate[pdf, {y, -Infinity, Infinity}];

iy = Integrate[y pdf, {y, -Infinity, Infinity}];

MeanOfyGivenx = m = (iy/i)

m = meany + rho (sy/sx) (x - meanx); (* Out below *)

iyy = Integrate[(y - m)^2 pdf, {y, -Infinity, Infinity}];

VarianceOfyGivenx = v = (iyy/i)

v = (1 - rho^2) sy^2;     (* Out below *)

Out = ConditionalExpression
[(meany sx + rho sy (-meanx + x))/sx,
Re[1/((-1 + rho^2) sy^2)] < 0]

Out = ConditionalExpression
[-(-1 + rho^2) sy^2,
Re[1/((-1 + rho^2) sy^2)] < 0]

CAVEAT: Six minutes elapsed time to evaluate on my PC

FranD

---- Ray Koopman wrote:

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.

---- Bob Hanlon wrote:

Clear[x, y];

mean1 = 58/10;
sigma1 = 2/10;

mean2 = 53/10;
sigma2 = 2/10;

rho = 6/10;

dist = MultinormalDistribution[{mean1, mean2},
{{sigma1^2, rho*sigma1*sigma2},
{rho*sigma1*sigma2, sigma2^2}}];

plot1 = Plot3D[PDF[dist, {x, y}],
{x, mean1 - 3 sigma1, mean1 + 3 sigma1},
{y, mean2 - 3 sigma2, mean2 + 3 sigma2},
PlotRange -> All]

m = Integrate[y*PDF[dist, {63/10, y}],
{y, -Infinity, Infinity}]

(14*Sqrt[2/Pi])/E^(25/8)

m // N

0.490792

Alternatively,

m == Expectation[y*DiracDelta[x - 63/10],
Distributed[{x, y}, dist]]

True

s = Sqrt[Integrate[(y - m)^2*PDF[dist, {63/10, y}],
{y, -Infinity, Infinity}]];

s // N

1.51329

Alternatively,

s == N[Sqrt[Expectation[(y - m)^2*DiracDelta[x - 63/10],
Distributed[{x, y}, dist]]]]

True

Bob Hanlon

---- Steve <s123 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

```

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