Condition 2: Proposition Two random variables and , forming a discrete random vector, are independent if and only if where is their joint probability mass function and and are their marginal probability mass functions. Let be a discrete random vector with support: In order to verify whether and are independent, we first need to derive the marginal probability mass functions of and.

Condition 1: Example Let the joint probability density function of and be Its marginals are and Verifying that is straightforward.

Two random variables are independent if they convey no information about each other and, as a consequence, receiving information about one of the two does not change our assessment of the probability distribution of the other. The above conditions are equivalent.

When and , then. This means that A and B are not independent. Table of contents Definition Independence criterion Independence between discrete random variables Independence between absolutely continuous random variables More details Mutually independent random variables Mutual independence via expectations Independence and zero covariance Independent random vectors Mutually independent random vectors Solved exercises Exercise 1 Exercise 2 Exercise 3.

Also the definition of mutual independence extends in a straightforward manner to random vectors.

Recall see the lecture entitled Independent events that two events and are independent if and only if. You can use tables like this to figure out whether two discrete random variables are independent or dependent. Thus, the probability mass function of is We need to compute the probability of each element of the support of: The converse is not true: All the equivalent conditions for the joint independence of a set of random variables see above apply with obvious modifications also to random vectors.

Definition We say that random vectors ,... AP Statistics: The support of is When , the marginal probability density function of is , while, when , the marginal probability density function of is Thus, summing up, the marginal probability density function of is The support of is When , the marginal probability density function of is , while, when , the marginal probability density function of is Thus, the marginal probability density function of is Verifying that is straightforward.

If two random variables and are independent, then their covariance is zero: A and B are independent random variables. Example Let be a discrete random vector with support Let its joint probability mass function be In order to verify whether and are independent, we first need to derive the marginal probability mass functions of and.

Let its joint probability mass function be Are and independent? Thus, the probability mass function of is The product of the marginal probability mass functions is which is obviously different from.