# Pairwise independence vs. mutual independence

#### StatOp

##### New Member
I'm doing a PhD in statistical optics and I've stumbled upon a set of random variables that may or may not be mutually independent. I've managed to show that they are pairwise independent, but in general that doesn't imply they're mutually independent. Is there some conditions which guarantees that a set of pairwise independent variables also are mutually independent? I need mutual independence in order to express an expectation of a product as the product of expectations.

#### Dason

Are you working with theoretical random variables or are you deriving this from data?

#### StatOp

##### New Member
Are you working with theoretical random variables or are you deriving this from data?
Theoretical random variables.

#### hlsmith

##### Omega Contributor
So, they have a potential unique function that generates them or you are assuming this. Do you know the data generating process/function?

I can grab the cost of milk and a national growth rate and they can be correlated and possibly dependent - so I think if you are not using data generated in a vacuum or on your computer there are always issues of possible correlations or if sample sizes are small type I error concerns. Not aware of a test, but I wouldn't be surprised if there was something. There are tests to see if data are randomly created (e.g., ? benford), so testing similarities without a deep contextual knowledge of source seems possible but not conclusive. Sorry if I just ramble here without any great help.

#### Dason

Can you give us more details then? Do you have the joint pdf/cdf?

#### StatOp

##### New Member
I'll state the problem later tonight with all known information. Is it possible to write Latex code here?

#### Dason

$$\frac{1}{2}$$