Calculating Covariance in Bivariate Normal Distribution - Step by Step Guide

[jimmy1]

I have 2 normally distributed dependent random variables, X and Y, and I have the mean and variance of both of them, and I want to find the covariance (or correlation) between X and Y.

Now the formula for the covariance is Cov(XY) = E(XY) - E(X)E(Y). So I tried to calculate E(XY) via the bivariate normal distribution, but it seems that to use the bivariate normal I need to provide the correlation coefficient as a parameter, but this is the parameter that I'm trying to actually find.

So how would I find an expression for the covariance of X and Y? To find E(XY), it seems you need to use P(XY), but to use this bivariate probability you need to provide the covariance (or correlation coefficient). So how do I get around this problem??

 

[EnumaElish]

You can numerically calculate the covariance by taking multiple observations from the two distributions and multiplying their values. The mean of their product is E(XY). For example, let X = daily humidity and Y = daily temperature. If I measure humidity and temperature over 100 days (or at 100 locations), I will have 100 ordered pairs of (X,Y), from which I can calculate E(XY).

 

[jimmy1]

Yes, with observations from the two distribitions I could calculate E(XY), but I need an expression for the covariance of X and Y, and not really an empirical result. That is given (X,Y) are both normally distributed with mean (m1, m2), and standard deviation (s1,s2) how do I find the covariance of XY, when they are dependent.

What really confuses me is that for all these bivariate distributions you need to supply a correlation coefficient or some sort of covariance parameter, yet there seems to be no way of actually obtaining these covariances for dependent variables?

So how does one actually use any of these bivariate distributions if it's not possible to theoretically get the covariances??

 

[EnumaElish]

If you think that two variables are jointly distributed but you only know the marginal distributions, the simplest way to obtain the joint dist. is to calculate the covariance empirically. Other than that, there are copulas: http://en.wikipedia.org/wiki/Copula_(statistics)

 

[mathman]

I have 2 normally distributed dependent random variables, X and Y, and I have the mean and variance of both of them, and I want to find the covariance (or correlation) between X and Y.

Now the formula for the covariance is Cov(XY) = E(XY) - E(X)E(Y). So I tried to calculate E(XY) via the bivariate normal distribution, but it seems that to use the bivariate normal I need to provide the correlation coefficient as a parameter, but this is the parameter that I'm trying to actually find.

So how would I find an expression for the covariance of X and Y? To find E(XY), it seems you need to use P(XY), but to use this bivariate probability you need to provide the covariance (or correlation coefficient). So how do I get around this problem??

When dealing with the theoretical distribution only (no empirical data), there is no way around the "problem". A bivariate normal distribution is defined by the covariance (or equivalent alternatives). If you don't have something, it remains undefined.

 

[jimmy1]

When dealing with the theoretical distribution only (no empirical data), there is no way around the "problem". A bivariate normal distribution is defined by the covariance (or equivalent alternatives). If you don't have something, it remains undefined.

So would I be right in concluding that when using any sort of multivariate distribution you have to know the correlation (covariance) between the random variables, and the only way to get these covariances is empirically?

 

[mathman]

So would I be right in concluding that when using any sort of multivariate distribution you have to know the correlation (covariance) between the random variables, and the only way to get these covariances is empirically?

Yes - unless there is some given theoretical information you can use.

 

[EnumaElish]

http://www.math.ethz.ch/%7Estrauman/preprints/pitfalls.pdf

Where not otherwise stated, we consider bivariate distributions of the random
vector (X; Y)^t.
Fallacy 1. Marginal distributions and correlation determine the joint distribution.
This is true if we restrict our attention to the multivariate normal distribution or
the elliptical distributions. For example, if we know that (X; Y)^t have a bivariate
normal distribution, then the expectations and variances of X and Y and the correlation
r(X; Y) uniquely determine the joint distribution. However, if we only know
the marginal distributions of X and Y and the correlation then there are many possible
bivariate distributions for (X; Y)^t. The distribution of (X; Y)^t is not uniquely
determined by F1, F2 and r(X; Y ). We illustrate this with examples, interesting in
their own right.