Question

Consider a circle of radius 1 with center at the point A = (1; 0). Let O denote the origin. A point B is chosen at random on the circle (not inside). Let X denote the X-coordinate of the point B. Find the pdf of X.

Answer 1

Consider the points on the circle in polar coordinates: all points on the circle are a fixed distance away from the center, and their coordinates are determined entirely by an angle $\theta\in[0,2\pi)$ made relative to the positive $x$-axis.

Any point B on the circle has $x$-coordinate $x(\theta)=1+\cos\theta$. Let $\Theta$ be a random variable uniformly distributed on the interval $[0,2\pi)$. Then

$P(\Theta=\theta)=\begin{cases}\frac1{2\pi}&\text{for }\theta\in[0,2\pi)\\0&\text{otherwise}\end{cases}$

$\implies P(\Theta\le\theta)=\begin{cases}0&\text{for }\theta$

Now,

$P(X\le x)=P(1+\cos\Theta\le x)=P(\Theta\le\cos^{-1}(x-1))$

so that the density of $X$ is

$P(X=x)=\dfrac{\mathrm d}{\mathrm dx}P(\Theta\le\cos^{-1}(x-1))$

$\implies P(X=x)=\begin{cases}\frac1{\pi\sqrt{2x-x^2}}&\text{for }0\le x\le2\\0&\text{otherwise}\end{cases}$