Re: MultinormalDistribution Question

*To*: mathgroup at smc.vnet.net*Subject*: [mg120291] Re: MultinormalDistribution Question*From*: Steve <s123 at epix.net>*Date*: Sun, 17 Jul 2011 06:02:34 -0400 (EDT)*References*: <201107100901.FAA24634@smc.vnet.net> <ivel76$89d$1@smc.vnet.net>

On Jul 11, 7:01 am, Andrzej Kozlowski <a... at mimuw.edu.pl> wrote: > First, your covariance matrix is not symmetric definite. I think you prob= ably meant: > > CapSigma = {{sigma1^2, rho*sigma1*sigma2}, {rho*sigma1*sigma2, > sigma2^2}} > > Assuming that, you can get the conditional expectation from the definitio= n: > > Integrate[y PDF[dist, {6.3, y}], {y, -Infinity, Infinity}]/ > PDF[MarginalDistribution[dist, 1], 6.3] > > 5.6 > > Alternatively, you can use Mathematica 8 built in NExpectation function: > > NExpectation[y \[Conditioned] 6.299 <= x <= 6.301, {x, y} \[Distribut= ed] dist] > 5.6 > > Once you have the conditional expectation, you can compute the conditiona= l variance, e.g. > > Chop[Integrate[(y - 5.6)^2*PDF[dist, {6.3, y}], {y, -Infinity, Infinit= y}]/PDF[MarginalDistribution[dist, 1], 6.3]] > 0.0256 > > Andrzej Kozlowski > > On 10 Jul 2011, at 11:01, Steve 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*sigma= 2} > > 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- Hide quoted text - > > - Show quoted text - Many thanks to all that responded to my question. I apologize for the errors in my original post. I hadn't yet figured out how to copy and paste Mathematica code into Usenet posts in a way that is human readable, I therefore hand-typed my original post which introduced typographical errors. Hopefully I can do better with this and future posts. The example problem of my original post came from a textbook on mathematical statistics where their solution is mean = 5.6 and stddev = 0.16 Andrzej Kozlowski's posting was particularly useful to me as it confirmed the textbook solution and showed how I might generalize this analysis to other situations. Below is my understanding of his solution. mean1 = 5.8; sigma1 = .2; mean2 = 5.3; sigma2 = .2; rho = .6; mu = {mean1, mean2}; capsigma = {{sigma1^2, rho*sigma1*sigma2}, {rho*sigma1*sigma2, sigma2^2}}; dist1 = MultinormalDistribution[mu, capsigma]; In[9]:= x1 = 6.3; conditionalpdf = PDF[dist1, {x1, y}]; a = Integrate[y*conditionalpdf, {y, -Infinity, Infinity}] ; marginaldistribution = MarginalDistribution[dist1, 1]; b = PDF[marginaldistribution, x1] ; mux2 = a/b; c = Integrate[(y - mux2)^2*PDF[dist1, {x1, y}], {y, -Infinity, Infinity}]; sigmax2 = Sqrt[Chop[c/b]]; {mux2, sigmax2} Out[18]= {5.6, 0.16} What I really need to do is perform this analysis on test data for which I have only a few data points, hence the Student T distribution would be more appropriate than the Normal distribution. Secondly, values for the "independent" and "dependent" variables have no physical meaning below zero. So this implies that I need truncated distributions. I'm hoping that the solution Andrzej provided can be generalized for these added complications. Here are my 9 {F,t} data points where "F" is considered "independent" and t considered "dependent". {{1.01041, 0.3152}, {10.455, 0.3386}, {17.9032, 0.2534}, {24.9581, 0.5412}, {26.4688, 0.3251}, {27.4651, 0.4428}, {30.1682, 0.3402}, {36.6174, 0.2106}, {45.6129, 0.2154}} Would someone be so kind as to plop this data into their notebook to confirm a solution or two for me ? My results are below which are based on truncating the Student T distribution, 8 degrees of freedom and a calculated rho of -0.2327. If F=1.0, the probability of t exceeding 0.5 is 18.24% If F=10.0, the probability of t exceeding 0.5 is 13.17% If F.0, the probability of t exceeding 0.5 is 9.24% If F=24.52, the probability of t exceeding 0.5 is 8.11% If F=30.0, the probability of t exceeding 0.5 is 7.20% If F=40.0, the probability of t exceeding 0.5 is 6.55% While working this, I came upon some integration problems and needed to increase MaxRecursion to 20 Thanks in advance, --Steve

**References**:**MultinormalDistribution Question***From:*Steve <s123@epix.net>