**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 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

**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 \(\pi\).

- Version: 22 November 2020
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