Consider the circle of radius $1$ centred at the origin in two dimensions. Choose two points $x$ and $y$ independently at random so that both are uniformly distributed on the circle. Let the vectors joining the origin to $x$ and $y$ be $X$ and $Y$, respectively. Let $\theta$ be the angle between $X$ and $Y$, measured in an anti-clockwise direction while moving along the circle from $x$ towards $y$. Which of the following is $\text{TRUE?}$
- $\mathbf{E}[\theta]=\pi$
- $\mathbf{E}\left[|x-y|^{2}\right]=\sqrt{2}$
- $\mathbf{E}\left[|x-y|^{2}\right]=1+\sqrt{2}$
- $\mathbf{E}\left[|x-y|^{2}\right]=\sqrt{3}$
- $\mathbf{E}\left[|x-y|^{2}\right]=1$