### Question 1 (16 marks)

- (i). Let
*X*_{1 }be an exponential random variable with rate parameter 1 and*X*_{2 }a Gamma random variable with parameter (2*,*1). Define*V*:=*X*_{1 }+*X*_{2}, determine the distribution of*V*. (3 marks)

(ii). Further define random variable *U *to be

*.*

Find the joint density function of *U *and *V *(you can use the transformation method), and hence show that *U *and *V *are independent. (5 marks)

- Briefly explain the following properties (no more than two sentences for each) of a continuous time Markov chain {
*X*(*t*) :*t*≥ 0} with states {0,1,2,…}. Note: you are allowed to utilise some mathematical formulas to help you explain (where applicable). (4 marks)- Markov property;
- Stationary transition probability;
- Independent increment.

- State whether the following stochastic processes are measured in discrete or continuous time with discrete or continuous state space. (e.g. Poisson process is measured in continuous time with discrete state space.) (4 marks)
- Branching process;
- Brownian bridge;
- Random walk;
- Birth and death process.

### Question 2 (20 marks)

Let {X}_{n}_{n}_{≥0 }be a Markov chain with state space S = {0,1,2,…,8} and transition |
|||||||||

matrix | |||||||||

0 0 1 0 0 0 0 1 |
0
0 0 1 0 0 0 0 0 |
0
1 0 0 0 0 0 0 0 |
0
0 1 0 0 0 0 0 0 |
0 0 0
0 3 1 0 0 0 |
0
0 0 0 0 0 0 3 0 |
0
0 0 1 0 0 1 0 0 |
0
0 0 1 1 0 0 1 0 |
1 3 0 0 0 0 0 0 0 |

- Provide a final accessibility diagram showing the communication classes. (3 marks)
- Classify each state as either transient, null recurrent, or positive recurrent. (3 marks)
- Find the hitting probabilities
*ρ*_{x,}_{0 }=**P**(_{x}*T*_{0 }*<*∞) for all*x*∈*S*. (4 marks) - Find the period of each state. (3 marks)
- Determine all stationary distributions. (5 marks)
- Does there exist a steady state distribution? Give reasons. (2 marks)

### Question 3 (10 marks)

An insurance company feels that a randomly chosen policyholder will make claims according to a conditional Poisson process with rate uniformly distributed over (0*,*2) and time measured in years.

- Derive the mean value and variance of the number of claims made by that policyholder in
*t*(6 marks) - Compute the probability that the policyholder makes exactly one claim in one year. (4 marks)

### Question 4 (12 marks)

Let {*W*(*t*) : *t *≥ 0} be standard Brownian motion. Introduce the Gaussian process {*X*(*t*) : *t *≥ 0}, defined by *X*(*t*) := 5*t *+ 3*W*(*t*). Let Φ(*z*) be the *cdf *of the standard normal distribution.

- Determine
**E**(*X*(*t*)) and*Cov*(*X*(*s*)*,X*(*t*)). (3 marks) - For all
*t >*0 write down the density of*X*(*t*). (3 marks) - Conditionally on
*X*(1) = 2, what is the probability that*X*(1*/*2)*> u*? Use*cdf*

Φ(*z*) of standard normal. Then, compute this probability when *u *= 1. (6 marks)

### Question 5 (18 marks)

A group of *n *orchids is being cultivated in a greenhouse. Suppose that each orchid plant flowers (creating a single orchid bloom) after an incubation time, *B _{i }*(

*i*= 1

*,…,n*), which is exponentially distributed with rate parameter

*λ*. Thus, we can use a finitestate pure birth process to model the flowering time of an individual orchid plant, with state 0 indicating that it has not yet flowered and state 1 indicating that it has flowered. Specifically, let

*X*(

_{i}*t*) indicate the state of the

*i*th orchid at time

*t*, so that

*X*_{1}(*t*)*,…,X _{n}*(

*t*) are independent processes on the state space

*S*= {0

*,*1} with transition functions given by:

*P*_{00}(*t*) = *e*^{−λt }and *P*_{11}(*t*) = 1*.*

Define the total flowering process, *Y *(*t*) = *X*_{1}(*t*) + *… *+ *X _{n}*(

*t*), which keeps track of how many of the

*n*orchids have flowered by time

*t*. It can be shown that

*Y*(

*t*) is itself a pure birth process on the state space

*S*= {0

_{Y }*,*1

*,…,n*}.

- What are the infinitesimal parameters of the process
*Y*(*t*)? Briefly interpret your result. [HINT: Consider the distribution of*τ*_{1}, the first transition time of*Y*(*t*).] (3 marks) - Determine the mean and variance of
*Y*(*T*), for some pre-specified fixed time*T*. (6 marks)

Orchid blooms are extremely beautiful, and therefore extremely valuable. Suppose that the worth of an orchid bloom is randomly distributed with mean $50.00 and standard deviation $10.00.

- Determine the mean and standard deviation of the total worth of a collection of
*n*= 100 orchid plants by the time*T*= 5, assuming*λ*= 0*.*[NOTE: You may assume that the worth of a bloom is independent of the time the orchid plant flowers.] (4 marks) - Now, suppose that we extend our model for individual orchid plants to allow for blooms dying after a random length of time,
*D*, which is exponentially distributed with rate_{i}*µ*and then the plant re-flowering later on in its life. In other words,*X*(_{i}*t*) is now a finite-state birth and death process on the state space*S*= {0*,*1} with infinitesimal parameters:

*q*_{00 }= −*λ*; *q*_{01 }= *λ*; *q*_{10 }= *µ*; *q*_{11 }= −*µ.*

As before, let *Y *(*t*) = *X*_{1}(*t*) + *… *+ *X _{n}*(

*t*), which now keeps track of how many of the

*n*orchids are in bloom at time

*t*. Find the expected long-run proportion of the

*n*orchids which will be blooming at any given time. (5 marks)

### Question 6 (14 marks)

The number of arrivals to a shop is governed by a Poisson process {*X*(*t*)*,t *∈ [0*,*8]}, with time dependent rate

*,*

- Derive the probability of no arrivals in the interval (3, 5]. (3 marks)
- Determine the expected values of the number of arrivals in the last 5 opening hours (i.e., in the interval (3, 8]), given that 15 customers have arrived in the last 3 opening hours (that is, in the interval (5, 8]). (5 marks)
- Given that 60 customers visited the shop during those 8 opening hours, find an approximate value to the probability that more than 40 of those 60 customers arrived in the interval (0
*,*6]. (6 marks)

### Question 7 (10 marks)

Two types of claims are made to an insurance company. Type I claims arrive by a Poisson process with rate 10 and each claim amount is exponentially distributed with mean $1500; Type II claims arrive by a Poisson process with rate 1 that is independent of the other process, and each claim amount is exponentially distributed with mean $4000. Let *α *be the discount rate (for example, a claim of 1000 that arrived at time *t *would have a discounted cost of 1000*e*^{−αt}).

- Write down an expression for the total discounted cost of all claims made up to time
*t*, defined as*D*(*t*). Thus, find the mean of*D*(*t*), as a function of*α*and*t*. (5 marks) - A claim of 2000 has just been received. What is the probability that it is a type

I claim? (5 marks)

### Question 8 (5 marks) (STAT7004 Only)

A symmetric random walk in two dimensions is defined to be a sequence of points{(*X _{n},Y_{n}*) :

*n*≥ 0} which evolves in the following way: if (

*X*) = (

_{n},Y_{n}*x,y*) then

*X*

_{n}_{+1}

*,Y*

_{n}_{+1 }is one of the four points (

*x*± 1

*,y*)

*,*(

*x,y*± 1), each being picked with equal probability . If

(*X*_{0}*,Y*_{0}) = (0*,*0). Show that **E **.