Problem source

A point $P$ is chosen at random (with uniform distribution) on the circle $x^{2}+y^{2}=1$. The random variable $X$ denotes the distance of $P$ from $(1,0)$. Find the mean and variance of $X$. 
Find also the probability that $X$ is greater than its mean. 

Solution source

Consider the angle from the origin, then $P = (\cos \theta, \sin \theta)$ where $\theta \sim U(0, 2\pi)$, and $X = \sqrt{(\cos \theta - 1)^2 + \sin^2 \theta}$

\begin{align*}
\mathbb{E}[X] &= \int_0^{2\pi} \sqrt{(\cos \theta - 1)^2 + \sin^2 \theta} \frac1{2\pi} \d \theta \\
&=  \frac1{2\pi}\int_0^{2\pi} \sqrt{2 - 2\cos \theta} \d \theta \\
&= \frac{1}{2\pi}\int_0^{2\pi} \sqrt{4\sin^2 \frac{\theta}{2}} \d \theta \\
&= \frac{1}{\pi}\int_0^{2\pi} \left |\sin \frac{\theta}{2} \right| \d \theta \\
&= \frac{1}{\pi} \left [ -2\cos \frac{\theta}{2} \right]_0^{2\pi} \\
&= \frac1{\pi} \l 2 + 2\r \\
&= \frac{4}{\pi}
\end{align*}
\begin{align*}
\mathbb{E}(X^2) &= \frac1{2\pi}\int_0^{2\pi} (\cos \theta - 1)^2 + \sin^2 \theta \d \theta \\
&= \frac1{2\pi}\int_0^{2\pi} 2 - 2 \cos \theta \d \theta \\
&= \frac{4\pi}{2\pi} \\
&= 2 \\
\end{align*}

$\Rightarrow$ $\mathrm{Var}(X) = \mathbb{E}(X^2) - \mathbb{E}(X)^2 = 2 - \frac{16}{\pi^2} = \frac{2\pi^2 - 16}{\pi^2}$.


\begin{center}
\begin{tikzpicture}[scale=2]

    \coordinate (O) at (0,0);
    \coordinate (P) at ({cos(2*asin(2/3.14159))},{sin(2*asin(2/3.14159))});
    \coordinate (X) at (1,0);
    \coordinate (PP) at ($(X)!0.5!(P)$);

    \draw (O) circle (1);

    \draw (-1.1, 0) -- (1.1,0);
    \draw (0,-1.1) -- (0,1.1);

    \draw (X) -- (P) -- (O);
    \draw[dashed] (O) -- (PP);

    \pic [draw, angle radius=0.8cm, "$\theta$"] {angle = X--O--P};

    \node at (PP) [right, above] {$\frac{4}{\pi}$};

\end{tikzpicture}
\end{center}

Where the line makes a length longer than $\frac{4}{\pi}$ it will make an angle at the origin of $2\sin^{-1} \frac{2}{\pi}$. Therefore the probability of being larger than this is $\frac{2\pi - 2 \times 2\sin^{-1} \frac{2}{\pi}}{2 \pi} = 1 - \frac{2}{\pi} \sin^{-1} \frac{2}{\pi} \approx 0.560$