5.6 Continuous independence
Independence
Two discrete RVs \(X\) and \(Y\) are independent if
\[
p_{X, Y}(x, y) = p_X(x) \cdot p_{Y}(y), \;\; \text{for all $x$ and $y$.}
\]
Two continuous RVs \(X\) and \(Y\) are independent if
\[
f_{X, Y}(x, y) = f_X(x) \cdot f_{Y}(y), \;\; \text{for all $x$ and $y$.}
\]
Let \((X, Y)\) be a random point on the unit square.
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Its joint PDF:
\[
\begin{aligned}
f_{X, Y}(x, y) &=
\begin{cases}
1, & \text{if $x, y \in [0, 1]$,} \\
0, & \text{otherwise.}
\end{cases}
\end{aligned}
\]
Let \((X, Y)\) be a random point on the unit square.
\[
\begin{aligned}
f_{X, Y}(x, y) &=
\begin{cases}
1, & \text{if $x, y \in [0, 1]$,} \\
0, & \text{otherwise.}
\end{cases}
\end{aligned}
\]
\[
\small{
\begin{aligned}
f_{X}(x) &=
\begin{cases}
1, & \text{if $x \in [0, 1]$,} \\
0, & \text{otherwise.}
\end{cases}
\end{aligned}
}
\]
\[
\small{
\begin{aligned}
f_{Y}(y) &=
\begin{cases}
1, & \text{if $y \in [0, 1]$,} \\
0, & \text{otherwise.}
\end{cases}
\end{aligned}
}
\]
\[
f_{X, Y}(x, y)=f_{X}(x) \cdot f_{Y}(y)
\]
\(X\) and \(Y\) are independent.
