1. The waiting time T between successive occurrences of an event E in a discrete time renewal process has the probability distribution

P(T =1) = 0.7, P(T = 2) = 0.3.

(iii) If an observer arrives after the renewal process has been running for a long time, what is the approximate probability that an event occurs at the next time point?

2. The waiting time T between successive occurrences of an event E in a discrete-time renewal process has the probability distribution

P(T =1) = 0.7, P(T = 2) = 0.3.

(ii) The waiting time to the sixth occurrence of E is denoted by W6. Find the probability P(W6 = 10).

3. An animal leaves its burrow near a river bank and moves up and down along the river bank foraging for food. Its distance upstream from its burrow after t minutes is denoted by X(t), and may be reasonably modeled as an ordinary Brownian motion {X(t); t ≥ 0} with diffusion coefficient σ2 = 3 (meters)2 per minute.

(i) Find the probability that after 3 minutes the animal is not further than 5 metres away from its burrow.

4. An animal leaves its burrow near a river bank and moves up and down along the river bank foraging for food. Its distance upstream from its burrow after t minutes is denoted by X(t), and may be reasonably modeled as an ordinary Brownian motion {X(t); t ≥ 0} with diffusion coefficient σ2 = 3 (meters)2 per minute.

(iii) If the animal is observed to be 30 meters upstream from its burrow after an hour, find the probability that it was downstream from its burrow after 20 minutes

5. An animal leaves its burrow near a river bank and moves up and down along the river bank foraging for food. Its distance upstream from its burrow after t minutes is denoted by X(t), and may be reasonably modeled as an ordinary Brownian motion {X(t); t ≥ 0} with diffusion coefficient σ2 = 3 (meters)2 per minute.

(ii) If the animal is observed to be 6 meters upstream from its burrow after 5 minutes, find the probability that the animal will be less than 2 meters upstream from the burrow after 15 minutes.