[next] [prev] [prev-tail] [tail] [up]

2.2 Independence

Theorem 2.2. If \(\underline {X}_1\) and \(\underline {X}_2\) are statistically independent then

1.
\(f\big (\underline {X}\big ) = f_1\big (\underline {X}_1\big )\cdot f_2\big (\underbrace {X}_2\big )\), where \(\underline {X} =\begin {pmatrix} \underline {X}'_1, & \underline {X}'_2\\ \end {pmatrix}\)
2.
\(f\Big (\underline {X}_2/\underline {X}_1\Big ) = f_2\big (\underline {X}_2\big )\) or \(f\Big (\underline {X}_1/\underline {X}_2\Big ) =f_1\big (\underline {X}_1\big )\)

Example 2.3. \(X_1\) and \(X_2\) in the previous example are dependent since \[f_1\big (X_1\big )f_2\big (X_2\big ) = 1 \neq 1 + (2x_1-1)(2x_2-1)\quad 0<x_1,x_2<1\quad -1\leq \alpha < 1\]

[next] [prev] [prev-tail] [front] [up]

Questions on this section

Stuck on something here? Ask below and it stays attached to this topic.