2102 Final Exam PASS Session
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Transcript of 2102 Final Exam PASS Session
ACTL2102 Final Exam PASS Session
ACTL2102 Final Exam PASS Session
October 30, 2014
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ACTL2102 Final Exam PASS Session
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
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ACTL2102 Final Exam PASS Session
Lectures 1-2: Stochastic Processes and Discrete Time Markov Chains
Stuff You Should Know
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
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ACTL2102 Final Exam PASS Session
Lectures 1-2: Stochastic Processes and Discrete Time Markov Chains
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.
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ACTL2102 Final Exam PASS Session
Lectures 1-2: Stochastic Processes and Discrete Time Markov Chains
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?
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ACTL2102 Final Exam PASS Session
Lectures 1-2: Stochastic Processes and Discrete Time Markov Chains
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
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ACTL2102 Final Exam PASS Session
Lectures 1-2: Stochastic Processes and Discrete Time Markov Chains
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.
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ACTL2102 Final Exam PASS Session
Lecture 3: Exponential Distribution and the Poisson Process
Stuff You Should Know
Stuff You Should Know
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
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ACTL2102 Final Exam PASS Session
Lecture 3: Exponential Distribution and the Poisson Process
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?
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ACTL2102 Final Exam PASS Session
Lecture 3: Exponential Distribution and the Poisson Process
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.
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ACTL2102 Final Exam PASS Session
Lecture 3: Exponential Distribution and the Poisson Process
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).
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ACTL2102 Final Exam PASS Session
Lecture 3: Exponential Distribution and the Poisson Process
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?
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ACTL2102 Final Exam PASS Session
Lecture 3: Exponential Distribution and the Poisson Process
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.
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ACTL2102 Final Exam PASS Session
Lectures 4-5: Continuous Time Markov Chains
Stuff You Should Know
Stuff You Should Know
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
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ACTL2102 Final Exam PASS Session
Lectures 4-5: Continuous Time Markov Chains
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.
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ACTL2102 Final Exam PASS Session
Lectures 4-5: Continuous Time Markov Chains
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
ACTL2102 Final Exam PASS Session
Lectures 4-5: Continuous Time Markov Chains
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 ].
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ACTL2102 Final Exam PASS Session
Lectures 4-5: Continuous Time Markov Chains
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.
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ACTL2102 Final Exam PASS Session
Lectures 4-5: Continuous Time Markov Chains
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.
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ACTL2102 Final Exam PASS Session
Lectures 6-8: Time Series Mathematics
Stuff You Should Know
Stuff You Should Know
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
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ACTL2102 Final Exam PASS Session
Lectures 6-8: Time Series Mathematics
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.
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ACTL2102 Final Exam PASS Session
Lectures 6-8: Time Series Mathematics
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).
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ACTL2102 Final Exam PASS Session
Lectures 6-8: Time Series Mathematics
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?
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ACTL2102 Final Exam PASS Session
Lectures 6-8: Time Series Mathematics
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 .
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ACTL2102 Final Exam PASS Session
Lectures 8-10: Model Selection, Checking, and Prediction
Stuff You Should Know
Stuff You Should Know
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
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ACTL2102 Final Exam PASS Session
Lectures 8-10: Model Selection, Checking, and Prediction
Example Questions
Mock Exam Q6 & Q7
No. I can’t be bothered typing them out. Just... open the .pdf foryourself.
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ACTL2102 Final Exam PASS Session
Lectures 8-10: Model Selection, Checking, and Prediction
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.
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ACTL2102 Final Exam PASS Session
Lectures 8-10: Model Selection, Checking, and Prediction
Example Questions
Tutorial 10-2 (Modified)
Explain briefly whether the following processes are Markov:
1 AR(4)
2 ARMA (1,1)
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ACTL2102 Final Exam PASS Session
Lectures 8-10: Model Selection, Checking, and Prediction
Example Questions
Tutorial 10-3 (Modified)
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.
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ACTL2102 Final Exam PASS Session
Lectures 8-10: Model Selection, Checking, and Prediction
Example Questions
Tutorial 10-4 (Modified)
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.
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ACTL2102 Final Exam PASS Session
Lectures 11-12: Pseudo-Random Continuous Walks
Stuff You Should Know
Stuff You Should Know
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
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ACTL2102 Final Exam PASS Session
Lectures 11-12: Pseudo-Random Continuous Walks
Example Questions
Lecture 11-30
Evaulate E[B8t ], where Bt is standard Brownian Motion.
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ACTL2102 Final Exam PASS Session
Lectures 11-12: Pseudo-Random Continuous Walks
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.
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ACTL2102 Final Exam PASS Session
Lectures 11-12: Pseudo-Random Continuous Walks
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.
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ACTL2102 Final Exam PASS Session
Lectures 11-12: Pseudo-Random Continuous Walks
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.
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ACTL2102 Final Exam PASS Session
Lectures 11-12: Pseudo-Random Continuous Walks
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.
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ACTL2102 Final Exam PASS Session
Lectures 11-12: Pseudo-Random Continuous Walks
Example Questions
WE’RE DONE HERE!
GOOD LUCK!
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