91Ó°ÊÓ

Let \(X\) and \(Y\) be independent variables having the exponential distribution with parameters \(\lambda\) and \(\mu\) respectively. Find the density function of \(X+Y\).

Three points \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) are chosen independently at random on the circumference of a circle. Let \(b(x)\) be the probability that at least one of the angles of the triangle ABC exceeds \(x \pi\). Show that $$ b(x)= \begin{cases}1-(3 x-1)^{2} & \text { if } \frac{1}{3} \leq x \leq \frac{1}{2} \\ 3(1-x)^{2} & \text { if } \frac{1}{2} \leq x \leq 1\end{cases} $$ Hence find the density and expectation of the largest angle in the triangle.

Lines are laid down independently at random on the plane, dividing it into polygons. Show that the average number of sides of this set of polygons is 4 . [Hint: Consider \(n\) random great circles of a sphere of radius \(R\); then let \(R\) and \(n\) increase.]