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.
![]()
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.
- We draw parallel lines with distance \(d\) from each other.
- We threw a needle of length \(l\) on the surface at random.
- What is the probability that the needle intersects a line?
- Let \(X\) be the vertical distance from the midpoint of the needle to the closest line.
- Let \(\Theta\) be the acute angle between the needle and lines.
![]()
\[
\begin{aligned}
f_{X}(x) &=
\begin{cases}
\frac{1}{d/2}, & \text{if $0 < x < \frac{d}{2}$,} \\
0, & \text{otherwise.}
\end{cases} \\
\\
f_{\Theta}(\theta) &=
\begin{cases}
\frac{1}{\pi/2}, & \text{if $0 < \theta < \frac{\pi}{2}$,} \\
0, & \text{otherwise.}
\end{cases} \\
\end{aligned}
\]
By independence of \(X\) and \(\Theta\), we have
\[
\begin{aligned}
f_{X, \Theta}(x, \theta) &= f_{X}(x) \cdot f_{\Theta}(\theta) \\
\\
&=
\begin{cases}
\frac{4}{\pi d}, & \text{if $0 < x < \frac{d}{2}, 0 < \theta < \frac{\pi}{2}$,} \\
0, & \text{otherwise.}
\end{cases}
\end{aligned}
\]
When does the needle intersect with a line?
The needle intersects with a line if and only if \(x < \frac{l}{2}\sin \theta\).
\[
\small{
\begin{aligned}
&\text{P}(\text{intersect}) \\
&= \text{P}(X < \frac{l}{2}\sin \Theta) \\
&=\iint \limits_{x < \frac{l}{2}\sin \theta} f_{X, \Theta}(x, \theta)dx d\theta \\
&=\iint \limits_{x < \frac{l}{2}\sin \theta} \frac{4}{\pi d}dx d\theta \\
&=\frac{4}{\pi d}\int _0^{\pi/2} \int _0^{\frac{l}{2}\sin \theta} dx d\theta \\
\end{aligned}
}
\]
\[
\small{
\begin{aligned}
&=\frac{4}{\pi d}\int _0^{\pi/2} \frac{l}{2}\sin \theta \; d\theta \\
&=\frac{2l}{\pi d}\int _0^{\pi/2}\sin \theta \; d\theta \\
&=\frac{2l}{\pi d} (-\cos \theta) \bigg |_0^{\pi/2} \\
&=\frac{2l}{\pi d} (0-(-1)) \\
&=\frac{2l}{\pi d} \\
\end{aligned}
}
\]
We can use this result to estimate \(\pi\)
To simplify, let \(l=d/2\).
\[
\text{P}(\text{intersect})=\frac{2l}{\pi d}=\frac{2(d/2)}{\pi d}=\frac{1}{\pi}
\]
\[
\text{P}(\text{intersect})\approx \frac{\text{# of intersections}}{\text{# of needles}}
\]
\[
\pi \approx \frac{\text{# of needles}}{\text{# of intersections}}
\]
\[
\pi \approx \frac{\text{# of needles}}{\text{# of intersections}}
\]
