2.5 Independence

The notion of independence is important.

Definition 1.1 (Independence) Two rvs \(X\) and \(Z\) are said to be independent if \(f_{X,Z}(x,z) = f_X(x) \times f_Z(z)\), where \(f_X(x)\) and \(f_Z(z)\) are the marginal distributions.
Example 2.10 (Independence) In Example 2.8, we see that \(f_{XZ}(x, z) \ne f_X(x) \times f_Z(z)\), so \(X\) and \(Z\) are not independent.
Example 2.11 (Independence) In Example 2.9, the permissable values of \(Y\) depend on the value of \(X\), so clearly \(X\) and \(Y\) are not independent.