Paper 2, Section II, F

A circular island has a volcano at its central point. During an eruption, lava flows from the mouth of the volcano and covers a sector with random angle $\Phi$ (measured in radians), whose line of symmetry makes a random angle $\Theta$ with some fixed compass bearing.

The variables $\Theta$ and $\Phi$ are independent. The probability density function of $\Theta$ is constant on $(0,2 \pi)$ and the probability density function of $\Phi$ is of the form $A(\pi-\phi / 2)$ where $0<\phi<2 \pi$, and $A$ is a constant.

(a) Find the value of $A$. Calculate the expected value and the variance of the sector angle $\Phi$. Explain briefly how you would simulate the random variable $\Phi$ using a uniformly distributed random variable $U$.

(b) $H_{1}$ and $H_{2}$ are two houses on the island which are collinear with the mouth of the volcano, but on different sides of it. Find

(i) the probability that $H_{1}$ is hit by the lava;

(ii) the probability that both $H_{1}$ and $H_{2}$ are hit by the lava;

(iii) the probability that $H_{2}$ is not hit by the lava given that $H_{1}$ is hit.