2102 final exam pass session

ACTL2102 Final Exam PASS Session


October 30, 2014

1/37


Plan

1 Lectures 1-2: Stochastic Processes and Discrete Time Markov ChainsStuff You Should KnowExample Questions

2 Lecture 3: Exponential Distribution and the Poisson ProcessStuff You Should KnowExample Questions

3 Lectures 4-5: Continuous Time Markov ChainsStuff You Should KnowExample Questions

4 Lectures 6-8: Time Series MathematicsStuff You Should KnowExample Questions

5 Lectures 8-10: Model Selection, Checking, and PredictionStuff You Should KnowExample Questions

6 Lectures 11-12: Pseudo-Random Continuous WalksStuff You Should KnowExample Questions

2/37


Lectures 1-2: Stochastic Processes and Discrete Time Markov Chains

Stuff You Should Know


Definitions:

Independent IncrementsStationary IncrementsMarkov Process

Transition Matrices

State Classifications

Mean Time in Transient States

Periodicity

Limiting Probabilities and their Conditions

Time Reversibility

Gambler’s Ruin

Branching Processes

3/37



Example Questions

PASS 3-4 (modified)

Specify the classes of the following Markov chain, and determinefor each class whether they are transient or recurrent:

1

13

23 0 0

12 0 1

2 00 0 1

434

0 0 12

12

2 Find the probability that the process will return to state 4

eventually if the process starts at state 4.

3 Find the expected number of times state 4 is visited if theprocess starts at state 4.

4/37



Example Questions

Tutorial 2-2

An unfeasibly large organisation has N employees. Each employeehas one one of three possible job classifications and changesclassification (independently) according to a Markov chain withtransition probabilities. 0.7 0.2 0.1

0.2 0.6 0.20.1 0.4 0.5

1 What percentage of employees are in each classification?

2 Is this process time-reversible?

5/37



Example Questions

PASS 4-4

For the following branching process, calculate the probability ofextinction when the number Y of offspring of each individualfollows the following distribution:

1 P(Y = 0) = 13 ,P(Y = 1) = 1

2 ,P(Y = 2) = 16

2 Y ∼ Bin(2, 0.6)

For both questions, assume the starting population X0 = 1

6/37



Example Questions

PASS 4-6

Han Solo moves among n + 1 star systems that are arranged in acircle. At each system it moves either to the next system in theclockwise direction with probability p or the counterclockwisedirection with probability q = 1− p. Starting at a specifiedsystem, call it system 0, find the probability that all star systemshave been visited before revisiting system 0.

7/37


Lecture 3: Exponential Distribution and the Poisson Process



Exponential Distribution: Properties and Results

Definition of Counting and Poisson Process

Interarrival and Waiting Times

Sum and Thinning of Poisson Process

Non-Homogenous and Compound Poisson

8/37



Example Questions

PASS 5-3

The average number of automobiles entering a mountain tunnelper minute period is 1. Excessive number of cars entering thetunnel during a brief period of time produces a hazardoussituation. Assuming the Poisson process:

1 Find the probability that the number of automobiles enteringthe tunnel during a 1 minute period exceeds 2?

2 What is the probability that the number of automobilesentering the tunnel will be less than 3 during a 3-minuteperiod?

3 What is the probability that no car enters the tunnel duringthe first 4 minutes?

9/37



Example Questions

PASS 5-6

An insurance portfolio contains policies for three categories ofpolicyholder: A, B and C. The number of claims in a year, N, onan individual policy follows a Poisson distribution with mean λ.Individual claim sizes are assumed to be exponentially distributedwith mean 4 and are independent from claim to claim. Thedistribution of λ, depending on the category of the policyholder, is:

Category Value of λ Proportion of policyholders

A 2 20%

B 3 60%

C 4 20%

Denote by S the total amount claimed by a policyholder in oneyear.

10/37



Example Questions

PASS 5-6 continued

1 Prove that E (S) = E [E (S |λ)].

2 Show that E (S |λ) = 4λ and Var(S |λ) = 32λ.

3 Calculate E (S).

4 Calculate Var(S).

11/37



Example Questions

Tutorial 3-6

Events occur according to a nonhomogenoous Poisson processwhose mean value function is given by

m(t) = t2 + 2t, t ≥ 0

What is the probability that n events occur between time t = 4and t = 5?

12/37



Example Questions

Tutorial 3-9

Insurer A has a combined home insurance and landlords insuranceportfolio. The total number of claims for this portfolio is modelledusing a Poisson process with expected claims 300 per year. Theproportion of landlords insurance claims was 1/5 of the overallclaims. Insurer A sells its home insurance portfolio to insurer B.Insurer B specialises in home insurance and has no landlordsinsurance policies. The expected number of claims for the oldportfolio of insurer B was 120.

1 Define the processes of the number of claims from insurer Afor both the home insurance portfolio and landlords insuranceportfolio before the takeover.

2 Define the processes of the number of claims of both insurerA and insurer B before and after the takeover of the homeinsurance by insurer B.

13/37


Lectures 4-5: Continuous Time Markov Chains



Definitions and First Principles

The ”Q” (Generator) Matrix and the embedded Markov Chain

Kolmogorov Equations and the ”P(s,t)” transition matrix

Limiting Probabilities

Time Reversibility

Birth-Death Processes (rates, structure, mean time,Kolmogorov equations, balance equations)

HSD models, and computing probabilities using integrals

(Maybe) Calculation methods

14/37



Example Questions

PASS 6-1

A 24 hour convenience store has three cashiers, Ralph, Vincent,Benjamin. Customers arrive according to a Poisson process withrate λ per hour and join the queue if there are already 3 customersserved. Each cashier serves customers with service timesexponential with mean 1

µ per hour.

1 Define a Markov chain to model the number of customers inthe store and write down the corresponding generator matrix(the instantaneous transition rate matrix).

2 Given that there is n (n > 3) customers in the store, determinean expression for the conditional probability that the numberof customers will remain unchanged over the next 15 minutes.

15/37



Example Questions

PASS 6-5 (modified)

Consider the following probability transition rate matrix with state1, 2, 3, 4:

Q =

−0.7 0.4 0.3 00.1 −1 0.3 0.60.1 1.9 −2.2 0.21.7 0.3 0.5 −2.5

1 Find the embedded probability matrix.

2 Find the probability that the fifteen transition will be intostate 3, given we started in state 1 and the thirteenthtransition was into state 2.

3 If possible, find the long-run proportion of time spent in eachstate.

4 Is this process time-reversible?16/37



Example Questions

PASS 7-1

Consider the birth and death process with λi = λ for i = 0, 1, 2, ...and µj = µ for j = 1, 2, 3, .... Denote Ti as the time spent in statei before moving to state i + 1. Find E[Ti ].

17/37



Example Questions

PASS 7-7

Consider a health (H), sickness (S) and death (D) model for anindividual aged x > 0 with the following rates:

The rate at which a healthy individual becomes sick is 0.001x

The rate at which a sick individual recovers is 0.002x

The rate at which a healthy individual dies is 0.001x

The rate at which a sick individual dies is 0.002x

1 Give an expression for the probability that a sick 65 year oldindividual stays sick for at least 1 year and then becomeshealthy and remain so till age 67.

2 Find the probability that a sick 65 year old individual remainssick until he dies.

18/37



Example Questions

Tutorial 5-1 (modified)

Consider two machines both with an exponential lifetime 1/λ.

1 There is a single repairdrone that can service machines at arate µ. Set up the Kolmogorov backward equations. Also, setup the forward equations.

2 Consider instead the case where the inevitable automation ofall our jobs has not yet occurred and the repairdrone isinstead a repairhuman. This repairhuman repairs at a rateµ(t), where t is the time since breakfast. Set up theKolmogorov backward and forward equations.

19/37


Lectures 6-8: Time Series Mathematics



Classical decomposition model

Removing deterministic seasonality and deterministic trends

Integrated Time Series

SARIMA models (and their subsets)

Causality and Invertibility

Calculating ACF:

Linear Filter MethodYule-Walker Equations

Calculating PACF

20/37



Example Questions

PASS 8-1

Find a filter of the form 1 + αB + βB2 + γB3 that passes lineartrends without distortion and that eliminates arbitrary seasonalcomponents of period 2.

21/37



Example Questions

PASS 10-6

Compute the ACF and the PACF for the following AR(2) processXt = 0.6Xt−1 + 0.2Xt−2 + Zt where Zt ∼W .N(0, σ2).

22/37



Example Questions

Tutorial 8-3

Consider the AR(2) process {Xt} satisfying:

Xt − φXt−1 − φ2Xt−2 = Zt

For what values ofφ is this a causal process?

23/37



Example Questions

UKCT6 10/07-10 (modified)

The time series Xt is assumed to be stationary and to follow anARMA(2,1) process defined by:

Xt = 1 +8

15Xt−1 −

1

15Xt−2 + Zt −

1

7Zt−1

1 Determine the roots of the characteristic polynomial, andexplain how their values relate to the stationarity of theprocess.

2 Find the ACF for lags 0, 1, and 2.

3 Determine the mean and variance of Xt .

24/37


Lectures 8-10: Model Selection, Checking, and Prediction



Model selection

Looking at picturesAIC, BIC (AKA SBC), AICc

Model parameter estimation (finding and using SACF andSPACF)

Residual Analysis: Portmanteau/Ljung-Box

Testing for non-stationarity

Looking at the diagramDickey-Fuller and Augmented Dickey-Fuller tests

Cointegrated Time Series

Markov Property

Forecasting

Box-JenkinsBest Linear Predictor

25/37



Example Questions

Mock Exam Q6 & Q7

No. I can’t be bothered typing them out. Just... open the .pdf foryourself.

26/37



Example Questions

PASS 12-5

Consider the following 2 time series:

Xt = 2t2 + 3 + Zt

Yt = 3(t − 1)2 + Zt

1 We say that (Xt ,Yt) are integrated of order d . Find d .

2 Are Xt and Yt cointegrated? If so, give the cointegrationvector. If not, give reasons why not.

27/37



Example Questions

Tutorial 10-2 (Modified)

Explain briefly whether the following processes are Markov:

1 AR(4)

2 ARMA (1,1)

28/37



Example Questions


Suppose {Xt} is a stationary time series with mean µ and ACFρ(.). Find the best linear predictor of Xn+h of the form aXn + b.

29/37



Example Questions


Consider an ARIMA(1,1,1) model.

1 Use the Box-Jenkins apporach to derive the one-step andtwo-step ahead forecasts, assuming the parameter values areknown.

2 Evaluate the prediction variance Var( Xn+1 − x̂n(1)), assumingthe parameter values are known.

30/37


Lectures 11-12: Pseudo-Random Continuous Walks



Definitions of Brownian Motion

Properties of Brownian Motion

Stochastic Differential Equations (!)

Stochastic Integrals (!)

Monte Carlo Simulation

Linear Congruential Formula

Inverse-Transform (Discrete and Continuous)

Accept-Reject (Discrete and Continuous)

Variance Reduction Techniques

Importance Sampling

Number of Simulations

31/37



Example Questions

Lecture 11-30

Evaulate E[B8t ], where Bt is standard Brownian Motion.

32/37



Example Questions

Bonus Question

Consider the geometric Brownian Motion

Y (t) = exp(φt + ψXt)

where Xt has the stochastic differential equation

dXt = µdt + σdBt

and where Bt is standard Brownian motion.Express dYt as a stochastic differential equation.

33/37



Example Questions

Another Bonus Question

1 Consider the linear congruential formula. Where a = 11,c = 37, m = 100 and x0 = 85, generate 3 random numberson U[0, 1].

2 Use these numbers to generate samplings from a Bin(3, 0.5)and an exponential distribution with mean 0.5.

3 If possible, use the exponential random variable samplesobtained to generate samplings from a Gamma(2, 3) randomvariable. Use the Acceptance-Rejection method.

34/37



Example Questions

PASS 13-5

Suppose you are simulating a set or normally distributed randomvariables with mean 65 and standard deviation 15. Find thenumber of simulations required so that you will be in a 0.5% bandof the true value, 95% of the time.

35/37



Example Questions

Lazy Questions:

1 Explain how using antithetical variates reduces estimatevariance.

2 Explain how using control variates reduces estimate variance.

3 Explain how importance sampling can increase computationalefficiency for determining expectations.

36/37



Example Questions

WE’RE DONE HERE!

GOOD LUCK!

37/37

2102 final exam pass session

Documents

stochastic processes

process timereversible

following markov chain

time series mathematicsstuff

following distribution

poisson processstuff

probability p

probability ofextinction