MAST30001 Stochastic Modelling
Tutorial Sheet 6
You’ve probably already seen/done some of these before, but it’s useful to do them
yourselves/see them again!
1. Let X ∼Exponential(λ), with λ > 0. Prove that P(X > s + t|X > t) = P(X > s)
for every s, t > 0.
2. Let (Xi)i∈N be independent random variables with Xi ∼Exponential(λi). Find the
distribution of Yn = mini≤nXi.
3. Let (Ti)i∈N be i.i.d. Exponential(λ) random variables, and let N be a Geometric(p)
random variable that is independent of the other variables.
(a) Find the moment generating function E[etT1 ] of T1.
(b) Let Y =
∑N
i=1Xi. Find the distribution of Y .
4. Let X ≥ 0 be a random variable satisfying
(∗) P(X > s+ t|X > t) = P(X > s), for all s, t ≥ 0.
Show that X ∼Exponential(λ), for some λ ≥ 0.