Independent variables

Two variables are independent if changing one has no effect on the other. Formally, their Joint distribution function is the product of individual probability density functions:

Two variables may be conditionally independent if they are independent only when given a third one:

where we're using conditional distribution functions. An example of conditional independence are the steps of a Markov chain.

Independence is a testable property. A chi-square test for independence can be used to do so.

Properties#

• The covariance is zero: $\text{cov}(X,Y)=0$. By extension, the correlation is also zero: $\rho=0$. Note that the converse does not hold. Covariance being zero is a necessary but not sufficient condition for independence.