{r setup, include=FALSE} knitr::opts_chunk$set(echo = TRUE)  {exercise, name="Integrals"} * Let$X \geq 0$be a continuous random variable with finite first moment. Prove that $$\mathbb{E} X = \int_0 ^{\infty} \mathbb{P}(X >t) dt = \int_0 ^{\infty}[ 1- F_X(t)]dt$$ Hint: use a double integral. * Let$X$and$Y$be nonnegative independent continuous random variables. Prove that for$t >0$, we have $$\mathbb{P}(XY > t) = \int_0 ^{\infty} \mathbb{P}(X >\tfrac{t}{y}) f_Y(y) dy,$$ where$f_Y$is the probability density function for$Y$. * Using the previous results prove that $$\mathbb{E}( X Y) = (\mathbb{E} X )(\mathbb{E} Y),$$ assuming all the expectations are finite.  {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)