Exercise 1 (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 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.

• 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 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.

• 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 $$\pi$$.