[Math] Random Variable Distance to Origin from Randomly Selected Point in the Unit Circle

chi squaredintegrationprobabilityrandom variables

Problem Statement:

For a sample space $S=\left \{ (x,y)|-1\leq x\leq 1,-1\leq y\leq 1
> \right \}$,

  • Sketch the sample space $S$.

Suppose a point is chosen at random from $S$. The random variable $R$
is defined as the distance from the origin of the random point if the
distance is less than 1, and $R$ is 2 if the distance from the origin
is greater than or equal to one.

  • Give a mathematical expression for the sample space for $R$, $S_{R}$. Sketch the sample space for $R$.

  • Find a mathematical expression for the equivalent event (call it $A^{'}$) in $S$ for the event $A=\left \{ R\leq r \right \}$, where
    $0<r<1$. Draw a sketch that shows $A^{'}$ as a subset of $S$ and find
    $P(A)$.

  • Find a mathematical expression for the equivalent event, $B^{'}$ in $S$ for the event $B=\left \{ R < 2 \right \}$. Sketch the event
    $B^{'}$ and find $P(B)$.

  • Find a mathematical expression for the equivalent event, $C^{'}$ in $S$ for the event $C=\left \{ R=2 \right \}$. Sketch the event $C^{'}$
    and find $P(C)$.

  • Find the distribution function for $R$, $F_{R}(r)$. Sketch a graph of this function.

So far, I've drawn the sample space $S$ as a unit square on the $x,y$ system. I've determined the mathematical expression for $R$ is:

$R=\begin{cases}
& \sqrt{x^2 + y^2} \text{ , } \sqrt{x^2 + y^2} < 1 \\
& 2 \text{ , } \sqrt{x^2 + y^2} \geq 1
\end{cases}$

and

$S_{R}=\left \{ r:0<r<1 \right \}$

Currently, I've set $A=\left \{ R \leq r \right \}, 0<r<1$ and $A^{'}=P[R \leq r], 0<r<1$

From other source online people have said the probability of selecting a point in the unit circle and whose distance is less than or equal to $r$ is $\frac{\pi r^2}{\pi}=r^2$.

I'm a little lost because I know you can have a form like:

$P\left \{ R \leq r \right \}=F_{R}(r)=\iint_{x^2+y^2 \leq r^2} f_{X,Y}(x,y)dxdy$

and that you can convert from rectangular to polar coordinates and use $rdrd\theta$. However, I'm not entirely sure what $f_{X,Y}$ is in these cases.

Best Answer

Well, what you have is the map $S\mapsto R$. $$(x,y)\mapsto \begin{cases}\sqrt{x^2+y^2} &:& 0\leq \sqrt{x^2+y^2}< 1 \\ 2 &:&1\leq \sqrt{x^2+y^2}\end{cases}$$

$R$ will be a mixed random variable.   It will have a probability density for $R\in[0;1)$ and a probability mass at $R=2$.   The support for $R$ is $[0\,;1)\cup\{2\}$

So for $0\leq r< 1$, if $A_r\equiv \{R\leq r\} $ then $A'_r\equiv \{x^2+y^2\leq r^2\}$

For the given boundary, $A'_r$ is basically the disc of radius $r$ around the centre, within the square $[-1;1]^2$.   Since $S$ has uniform density and area 4, then $\mathsf P(A'_r) = \frac 14\pi r^2$.

And so forth...

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