Independence

Independence of Random Variables

For a sequence of RV’s $X_1,X_2,\dots X_n$there are two principal notions of independence:

• Mutual independence: independence property satisfied for all subsets of ${X_i}_{1<i<n}$. This is the notion of absolute independence we are often interested in.
• Pairwise independence: independence property satisfied only for pairs.

Independence of variates

It’s impossible to test for independence without the joint probability distribution, so in the context of statistics, and more specifically Forecasting, we will test for consequences of independence like uncorrelatedness ie just the lack of a linear relationship, not the lack of any relationship at all. We know correlation → dependence so we will run a blood test to make sure nothing is seriously wrong.