Hi!I have a set of normal distribution N(a1, s1), N(a2,s2), N(a3, s3) , .... where ai and si (i=1, ..) are known.I dont know if they are or not independents and I don't know how is his joint distribution (but probably is a normal distribution too).Are any way to know if the normal distributions are independent from each other??

For a normal distribution, the only condition that needs to be satisfied for independence is that the covariance matrix is diagonal.

If you don't know the joint dist. then you can find rvs that are both normal while being independent or dependant.e.g. let rv X~N(0,1) and another independent rv Y~N(0,1) clearly satisfies your info., ie both nromal and independent by definition.Alternatively let rv X~N(0,1) and let Y=X+1 ~ N(1,1) again they are both normally distributed but this time Y is clearly dependant on X.

Ramsey:I think that I dont understand your reply.Do you want to mean that when I find the mean and desviation of all my random variables i could work with it like there were independent? PS: Thanks for your help Ramsey

No.From your 2nd post it seems you just want to know if the sum of normal rvs is also normal. The answer is yes as long as the rvs are jonintly normally distributed.e.g if X and Y are jointly normal with covariance Cov(X,Y) (which includes independent X and Y) then the X+Y is also normal with mean {E(x) + E(Y)} and variance {Var(x)+2Cov(X,Y)+Var(Y)} . Try to generalise when there are more than two summation terms.

Well I disagree with our point (1.)Suppose X has a normal distribution with expected value 0 and variance 1. Let W = 1 or −1, each with probability 1/2, and assume W is independent of X. Let Y = WX. Then I claim that:Cov(X,Y) = 0; X and Y have normal distributions; and X and Y are not independent!If you don't know the joint distribution is multinormal you don't know the dependence structure so knowing the cov will not help.

Yes, my mistake, Ramsey is correct.However, if you are generating the normally distributed random variables seperately, then they are independent by construction, no?

Yes, the sum of independent normal rvs is also normal.

You didn't clarify whether `the normal distributions are all related to one underlying' or you're trying to model `the joint distribution of multiple underlyings each with a marginal normal distribution'. These are two different questions.If the normal distributions are on one underlying, as for example in Kalman filtering where you need to combine the result of two independent measurements with distributions and , the combination of the two normals is also normal with its mean and inverse variance given by , . If the normal distributions are over two different underlyings you need to use the copula method to construct their joint distribution. For example in the bivariate case the copula function is where F_1 and F_2 are univariate and bivariate cumulative normal distributions, and you also need to have their correlation coefficient or calculate the sample correlation from the data.

If you have say 2 N distributions then you have state whether or not they are correlated or not or can be arbitrary. If the correlation matrix is given then of course you attempt to transform it to the diagonal type. This is from probability theory point of view.In math statistics you have two samples. There exist tests for testing normality and independence. Based on numerical results you need to take a decision in statistical framework. This is pair wise procedure.

Thanks guys!I think that maybe I will have to use a copula method if I want to go in that way.

I'm not sure I understand the problem, but if I do you're going about it the wrong way.I think you have a computer program that gives you draws from several different Normal distributions with known means and variances. You want to know if you sum the numbers from each draw the result will have a Normal distribution. I'm not sure why you can't look at the code to see if the individual variables are independent, I assume it's either too complicated or you don't have access to the source code.Anyway, it's silly to worry about covariance. Test what you care about, not something that might be mathematically equivalent. In mathematics, if you know you have a draw from a multivariate Normal distribution, you know the sum of the variable (or any linear combination) has a Normal distribution. That's logic. But if you test and cannot reject that the draw comes from a multivariate Normal distribuiton, it does not mean that the sum is close enough to Normal for your purposes.If you want to know if the sum is Normal, add up the number from some draws and use any standard test for Normality. More likely, you don't really care if the result is Normal but only some feature of the distribution, like symmetry or thin tails or a 1% point 2.33 standard deviations from the mean. Whatever the feature is, test for it directly.

raul85- As far as I could figure it out that either u r talking about how to generate dependent normal variables or how to estimate the correlation matrix from a given sample...later is very subjective..that is what kind of estimate you want..that is MLE,unbiased etc etcformer one is as discussed below :copula is a solution..but correlation structure here you have to assume ... for gaussian copula u need to multiply cholesky matrix of correlation matrix to vector of (x1,x2,...xn) where xi are iid normal.I think this helps.thanx