```{exercise, name="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, name="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. * By conditioning on the first arrival, show that $$\mathbb{P}(E(t) >y) = \int_0 ^t \mathbb{P}(E (t-x) >y)dF(x) + \int_{t+y} ^{\infty} dF(x)$$ * Apply the general theorem on renewal equations to obtain that $$ \mathbb{P}(E(t) \leq y) = F(t+y) - \int_0 ^t [1 - F(t+y -x)] dm(x).$$ * Assuming the inter-arrivals are non-lattice type, apply the key renewal theorem to obtain that $$\lim_{t \to \infty} \mathbb{P}(E(t) \leq y) = \frac{1}{\mu} \int_0 ^y [1-F(x)]dx.$$ ```

```{exercise, name="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. * Suppose that I pick a large $t$, is it equally likely that it would be covered the tile types? * Run a simulation to estimate the probability that $t$ is covered by tile of length $\pi$. ```

* Version: `r format(Sys.time(), '%d %B %Y')` * [Rmd Source](https://tsoo-math.github.io/ucl/QHW6.1.Rmd)