CS计算机代考程序代写 MAST30001 Stochastic Modelling

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.