2.5 Independence

The notion of independence is important.

Definition 2.1 (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.