Stochastic Independence

Read HC 2.4 up to the end of Example 4 and note the definition of stochastically independent random variables (HC Definition 2). The word stochastic is often omitted.

The case of mutual independence for more than 2 variables is summarized below. Definition 2.5 gives an alternative criterion in terms of CDF's.

Definition 2..4

Let $(X_1,\,X_2,\, \dots ,\, X_n)$ be an $n$-dimensional continuous random vector with joint pdf $f_{X_1,\,\dots ,\,X_n}(x_1,\, \dots ,\,x_n)$ and range space ${\cal R}_n$. Then $X_1,\,X_2,\,\dots ,\, X_n$ are defined to be stochastically independent if and only if

$$f_{X_1,\,\dots ,\,X_n}(x_1,\,\dots ,\,x_n)=f_{X_1}(x_1)\dots f_{X_n}(x_n)$$ (2.7)

for all $(x_1,\,\dots ,\,x_n) \in {\cal R}_n$.

Definition 2..5

Let $(X_1,\,X_2,\, \dots ,\, X_n)$ be an $n$-dimensional random vector with joint cdf
$F_{X_1,\,\dots ,\,X_n}(x_1,\,\dots ,\,x_n)$. Then $X_1,\,X_2,\dots ,\,X_n$ are defined to be stochastically independent if and only if

$$F_{X_1,\,\dots ,\,X_n}(x_1,\,\dots ,\,x_n)=F_{X_1}(x_1)\dots F_{X_n}(x_n)$$ (2.8)

for all $x_i$.

Comments

HC's Theorem 1 on page 102 says, in effect,

1. If the joint pdf of $X_1,\,\dots ,\,X_n$ factorizes into $g_1(x_1)\dots g_n(x_n)$, where $g_i(x_i)$ is a function of $x_i$ alone (including the range space), $i=1,\,2,\,\dots ,\,n$, then $X_1,\,X_2,\,\dots \,X_n$ are mutually stochastically independent. It is not assumed that $g_i(x_i)$ is the marginal pdf of $X_i$.
2. Similarly, if the joint cdf of $X_1,\,\dots ,\,X_n$ factorizes into $G_1(x_1)\dots G_n(x_n)$ where $G_i(x_i)$ is a function of $x_i$ alone, then $X_1,\,\dots ,\,X_n$ are mutually stochastically independent.