Exercise 1 (Integrals)

Exercise 2 (Renewal equations) Refering the general theorem on renewal equations, show that if \(m\) be a renewal function with \(F\) as the cumulative distribution for the inter-arrival times, and \[ \phi = H + H*m,\] then \(\phi\) satisfies the renewal-type equation \[ \phi = H + \phi*F.\]

Exercise 3 (Excess life) With the usual notation, let \(E\) be the excess life of a renewal process with renewal function \(m\) and \(F\) for the cumulative distribution of the inter-arrival times.

\[ \mathbb{P}(E(t) \leq y) = F(t+y) - \int_0 ^t [1 - F(t+y -x)] dm(x).\]

\[\lim_{t \to \infty} \mathbb{P}(E(t) \leq y) = \frac{1}{\mu} \int_0 ^y [1-F(x)]dx.\]

Exercise 4 (Random tiles) I have two types of tiles, one of length \(\pi\) and another of length \(\sqrt{2}\). Suppose that I tile the half line \([0, \infty)\), via the following procedure, I pick one of two types of tiles with equal probability, then I can place it, starting at the origin. I continue this procedure indefinitly, and independently.