Subject CS2 Revision Notes For the 2021 exams Stochastic process models Booklet 1 Covering Chapter1 Stochastic processes The Actuarial Education CompanyPAST EXAM QUESTIONS This section contains all of the past exam questions from 2010 to 2018, plus those from the 2019 Paper A exams, relating to the topics covered in this booklet. Solutions are given after the questions. These give enough information for you to check your answer, including working, and also show you what an outline examination answer should look like. Further information may be available in the Examiners’ Report, ASET or Course Notes. (ASET can be ordered from ActEd.) We first provide you with a cross-reference grid that indicates the main subject areas of each exam question. You can use this, if you wish, to select the questions that relate just to those aspects of the topic that you may be particularly interested in reviewing. Alternatively, you can choose to ignore the grid, and attempt each question without having any clues as to its content. Page 14 © IFE: 2021 Examinationspajdweye uojysenD asiou ayy See0OKT duun{ nowew ‘ujeUp AOWeW, NEM WopUueY sseooid Bujunog ‘sse001d uossiog punodwios ‘'sse00/d UOSSIOg sjuswaoul Aueuoners sujed ajdwes eoeds areys 12s OUILL Cross-reference grid worsen Page 15 v 10 1 12 13 14 15 16 17 18 19 © IFE: 2021 ExaminationsSubject CT4 April 2010 Question 3 For each of the following processes: * counting process * general random walk compound Poisson process * Poisson process © Markov jump chain (a) state whether the state space is discrete, continuous or can be either (b) state whether the time set is discrete, continuous, or can be either. [5] Subject CT4 April 2011 Question 7 ()_ Define a counting process. 2] (i) For each of the following processes: * simple random walk * compound Poisson © Markov chain (a) state whether each of the state space and the time set is discrete, continuous or can be either (b) give an example of an application which may be useful to a shopkeeper selling dried fruit and nuts loose. {6} [Total 8] Subject CT4 September 2011 Question 3 Describe how a strictly stationary stochastic process differs from a weakly stationary stochastic process. GB Page 16 © IFE: 2021 ExaminationsSubject CT4 April 2012 Question 1 ()_ Define a general random walk. t (ii) State the conditions under which a general random walk would become a simple random walk. (1) [Total 2] Subject CT4 April 2013 Question 3 For both of the following sets of four stochastic processes, place each process in a separate cell of the following table, so that each cell correctly describes the state space and the time space of the process placed in it. Within each set, all four processes should be placed in the table: Time space. Discrete Continuous Discrete ite Space & Continuous (a) General random walk, compound Poisson process, counting process, Poisson process. (0) Simple random walk, compound Poisson process, counting process, white noise. 6) © IFE: 2021 Examinations Page 17Subject CT4 September 2013 Question 3 @ Define a Poisson process. 2 A bus route in a large town has one bus scheduled every 15 minutes. Traffic conditions in the town are such that the arrival times of buses at a particular bus stop may be assumed to follow a Poisson process. Mr Bean arrives at the bus stop at 12 midday to find no bus at the stop. He intends to get on the first bus to arrive. (i) Determine the probability that the first bus will not have arrived by 1:00pm the same day. 2] The first bus arrived at 1:10pm but was full, so Mr Bean was unable to board it. (iii) Explain how much longer Mr Bean can expect to wait for the second bus to arrive. 1) (iv) Calculate the probability that at least two more buses will arrive between 4:10pm and 1:20pm. (2) [Total 7] Subject CT4 September 2014 Question 1 For each of the following processes: ‘* counting process © simple random walk * compound Poisson process © Markov jump process (state whether the state space is discrete, continuous or can be either [2] (ii) state whether the time set is discrete, continuous or can be either. [2] [Total 4] Page 18 © IFE: 2021 ExaminationsSubject CT4 September 2014 Question 7 (i) Define a Poisson process. 2] (ii) Prove the memoryless property of the exponential distribution. (2) ‘Suppose there are three independent exponential distributions: X with parameter x Y with parameter y Z with parameter z (iii) (a) Demonstrate that min(X,¥,Z) is also an exponential distribution. (b) Give the parameter of this exponential distribution. 2 The arrivals of different types of vehicles at a toll bridge are assumed to follow Poisson processes whereby: Type of vehicle Rate Motorcycle 2 per minute Car 5 per minute Goods vehicle 1.5 per minute The toll for a motorcycle is £1, for a car £2 and for a goods vehicle £5. (iv) State the name of the stochastic process that describes the total value of tolls collected. 11) (v) Calculate the expected value of tolls collected per hour. (1) On the advice of a structural engineer, no more than two goods vehicles are allowed across the bridge in any given minute. If more than two goods vehicles arrive then some goods vehicles have to wait to go across. (vi) Calculate the probability that more than two goods vehicles arrive in any given minute. 2 (vii) Calculate the probability that exactly £4 in tolls is collected in a given minute. [4] [Total 14] © IFE: 2021 Examinations Page 19Subject CT4 April 2015 Question 1 For a simple random walk: () Define the process. (2) (i) Write down the nature of the state space and time space in which it operates. t) (iii) Describe an example of a practical application of the process. 1 [Total 4] Subject CT4 September 2015 Question 5 (part) The following diagrams illustrate sample paths for four stochastic processes. sneha ay i I 3} ee 7 ~ Tine (i) Identify which sample path is most likely to correspond to a: © discrete time, discrete state process * continuous time, discrete state process e discrete time, continuous state process * continuous time, continuous state process. [2] Page 20 © IFE: 2021 Examinations1 12 Subject CT4 April 2016 Question 5 () Define the following types of stochastic process: (a) a Poisson process (b) a compound Poisson process G3) Consider the modelling of the following situations: A. the number of claims for motorcycle accidents received by an insurer’s telephone claim line B the number of breakfast bagels sold by a New York bagel bar C the number of breakdowns of freezers in a large supermarket D the cost of wasted food caused by breakdowns of freezers in a large supermarket. (ii) Comment on which of the following stochastic processes will be most ‘suitable for modelling each of the four situations above: « — time-homogeneous Poisson process « time-inhomogeneous Poisson process * time-homogeneous compound Poisson process * time-inhomogeneous compound Poisson process [6] [Total 9] Subject CT4 April 2017 Question 2 () Define an increment of a process. t The rate of mortality in a certain population at ages over exact age 30 years, h(30 +u),, is described by the process: (30 +u)=B(1+y)" u20 where B and » are constants. (ii) Show that the increments of the process log[h(30 + u)] are stationary. [3] [Total 4] © IFE: 2021 Examinations Page 2113 14 15 Subject CT4 April 2017 Question 4 () Describe how a classification based on the nature of the state and time spaces of stochastic processes leads to a four-way categorisation. [2] (i) List FOUR stochastic processes, one for each of the four categories in your answer to part (i). [2] [Total 4] Subject CT4 September 2017 Question 2 For each of the following processes: ‘© general random walk Markov jump process * compound Poisson process Markov chain (a) state whether the state space is discrete, continuous or can be either (b) state whether the time set is discrete, continuous, or canbe either. [4] Subject CT4 September 2017 Question 3 Calls arrive on Fred’s desk phone according to a Poisson process with parameter 3, with time measured in hours. () Write down the expected number of phone calls Fred receives each hour. 1 Fred has not received a phone call for 15 minutes. (i) Give the expected time until Fred next receives a phone call (1) Fred goes into a meeting for half an hour. (iii) Determine the probability that Fred has NOT missed a call when he retums to his desk. t) The average length of a calll to Fred is 7 minutes. (iv) Determine the probability that if a caller phones Fred the line will be engaged, assuming that Fred is at his desk to receive calls. (2) [Total 5] Page 22 © IFE: 2021 Examinations16 17 Subject CT4 April 2018 Question 2 A football match between two teams, Team A and Team B, is being decided by a penalty competition. Each team takes one penalty alternately. Team A goes first. Let X; be the total number of penalties scored by Team A minus the total number of penalties scored by Team B afer the / th penalty has been taken. If X; =2, Team A wins and the competition stops. If X; = -2, Team B wins and the competition stops. (i) Determine the possible sample paths for the process X; for (=12,3,4. 13] the chance of Team A scoring each of its penalties is 0.5, and the chance of Team B scoring each of its penalties is 0.4. (i) Determine the distribution of X, for i=2 and /=3. 3) [Total 6] Subject CT4 April 2018 Question 4 (part, adapted) (Describe what is meant by the following terms: (a) discrete state space (b) stochastic process (©) continuous time model (d)_stochastic process of mixed type (4) © IFE: 2021 Examinations Page 2318 19 20 Subject CT4 September 2018 Question 3 For each of the following processes: * simple random walk * Markov jump process ‘* compound Poisson process Markov chain ‘* counting process (a) state whether the state space is discrete, continuous, or can be either (b) state whether the time set is discrete, continuous, or canbe either. [5] Subject CS2 April 2019 Question 4 (Define a general random walk 2 (i) Show that a general random walk is not stationary. BI [Total 5] Subject CS2 September 2019 Question 3 Astochastic process X; is defined as follows, over time points t = 0,1,2.... Xq=0 P[X.4-% =] =P P[X n= X=-i]=1-p (State the name, the state space and the time domain of this process. [2] (i) Write down the distribution of this process for f = 1,2 and 3. BI (ii) State the distribution of the process after a large number of time Periods N , including its parameters in terms of N and p t [Total 6] Page 24 © IFE: 2021 ExaminationsSOLUTIONS TO PAST EXAM QUESTIONS The solutions presented here are just outline solutions for you to use to check your answers. See ASET for full solutions. Subject CT4 April 2010 Question 3 State spaces and time sets Accounting process has discrete states but its time set can be either discrete or continuous. A general random walk may have either discrete or continuous states but its time set is discrete. ‘A compound Poisson process may have either discrete or continuous states but its time set is continuous. A Poisson process has discrete states and its time set is continuous. A Markov jump chain has discrete states and its time set is discrete. Subject CT4 April 2011 Question 7 () Counting process A counting process can have either a discrete or a continuous time set. It has discrete states 0,1,2,3, Its value is a non-decreasing function of time, which means that it can never drop back to a lower state. (ii)(@) State spaces and time sets Assimple random walk has discrete states and a discrete time set. A compound Poisson process may have either discrete or continuous states but its time set is continuous. A Markov chain has discrete states and its time set is discrete. © IFE: 2021 Examinations Page 25(i)(b) Examples of applications Examples of a simple random walk include: * the number of five pound notes in the till each time a five pound note passes between the shopkeeper and a customer * the number of days so far this year that the shopkeeper has made a profit minus the number of days he or she has made a loss. Examples of a compound Poisson process include: ‘* the total number of bags used in sales by time t ‘* the total weight of nuts sold by time f. Examples of a Markov chain include: ‘* the number of people in the shop each time the doors are opened * the no claims discount level on the shopkeepers delivery van. Subject CT4 September 2011 Question 3 Strictly stationary versus weakly stationary processes A stochastic process is stationary if its statistical properties do not change ‘over time. Expressed mathematically, a process {X;} is strictly stationary if, for any positive integer 1, the joint distribution of (X,,,%;,.-...X;,) and (Xpoke Xtpaso-Xtzan) is the same for all values of k , ie if all the statistical properties remain the same when the times involved are all shifted by the same amount. This is a more stringent condition than is usually required in practice. More often, weak stationarity is used, which only assumes that the first two moments of the process (ie the means, variances and covariances) remain constant over time. Page 26 © IFE: 2021 ExaminationsA process {X;} is weakly stationary if the following two conditions hold: © E(X;)=E(X;,,) forall t and k , je if the mean value is the same at all times © — cov(X;,X},4) depends only on k, ie if the covariances depend only on the ‘lag’ k . Subject CT4 April 2012 Question 1 ()_ Define a general random walk If Y;,,¥2,... are independent and identically distributed random variables, then the process X,, = Y; +Y> +--+ Yq (with Xq = 0) is a general random walk. (i) Conditions for a simple random walk If ¥j,¥,... (the ‘steps’ in the walk) can only take the value —1 or 1, it is a simple random walk. Subject CT4 April 2013 Question 3 Set (a) The only combination that will work here is: Time Set Discrete Continuous 5 Discrete Counting process Poisson process 2 . General random Compound Poisson & | Continuous walk process © IFE: 2021 Examinations Page 27The logic here is that we have no choice for where to put the Poisson process, but there is some flexibility with the other three. The general random walk and compound Poisson process have discrete and continuous time sets respectively. We need another discrete time set. So we must choose the discrete option for the time set of the counting process, which then forces it to go in the top left box. This then forces us to take the continuous state space option for the remaining two processes, which go in the bottom row. Set (b) The only combination that will work here is: Time Set Discrete Continuous 8 Discrete ‘Simple random Counting process & walk 2 ; - a Compound Poisson & Continuous White noise isa The logic this time is that we have no choice for where to put the simple random walk, which must go in the top left box. The counting process has a discrete state space, so this must go in the top right box. The compound Poisson process can then only go in the bottom right box (since it has a continuous time set) and the white noise must go in the bottom left. Subject CT4 September 2013 Question 3 (Define a Poisson process See Core Reading Paragraph 27. (i) Probability that the first bus will not have arrived by 1pm The probability that the first bus will not have arrived by 1pm is the probability that no buses arrive (ie no events occur) during the 1 hour period between midday and 1pm. We are told that the buses are scheduled to arrive once every 15 minutes. So 4= 4 (if we work in minutes). Page 28 © IFE: 2021 ExaminationsThe number of events in one hour has a Poisson distribution with mean: =i eos At=s)= £x60=4 So the probability of no events occurring is: e* =0.0183 (iii) How long can he expect to wait for the second bus? Since Poisson processes are memoryless, the expected time till the next event will always have the same distribution, T ~ Exp(4). So the time he can expect to wait till the next bus is E[7] =15 minutes. (iv) Probability of at least two more buses The number of buses arriving between times s and t has a Poisson|[A(t-s)] distribution. So the number of buses arriving in a 10-minute period has a Poisson| = x10]=Poisson[ 2] distribution. So the probability of at least two more buses arriving in the next 10 minutes is: Subject CT4 September 2014 Question 1 ‘A counting process has discrete states but its time set can be either discrete or continuous. A simple random walk has discrete states and its time set is discrete. A compound Poisson process may have either discrete or continuous states but its time set is continuous. ‘A Markov jump process has discrete states and its time set is continuous. © IFE: 2021 Examinations Page 29Subject CT4 September 2014 Question 7 (Define a Poisson process A Poisson process with rate 4 is a continuous-time process N; with discrete state space {0,1,2,...} for which: © No=0 * — N; has independent, stationary increments © N,-N, ~Poisson[a(t-s)], t>s. (ii) Prove the memoryless property Let T denote the waiting time till the next event in a Poisson process with rate 2. Then T ~ Exp(2) and the memoryless property states that, for any s>0, P(T >t+s|T >s)=P(T >t). We can prove this as follows: P(T>t+sT>s) PIT >s) _ en Altes) 7 PUT >t+s|T>s)= is e =P(T >t) (i)(a)— min(X,Y,2) We can demonstrate this by considering the distribution function of min(X,Y,Z) : P[min(X,¥,Z) < t]=1- P[min(X, Y,Z) >t] =1-P[X>tY>tZ>t] -P(X > t)P(Y > 1)P(Z>t) (using independence) ata entlxryt2) This matches the distribution function of an exponential distribution. Page 30 © IFE: 2021 Examinations(iii)(b) Parameter of the distribution The parameter for the distribution is x +y +z. (iv) Name of the stochastic process The total value of tolls collected form a compound Poisson process. (v) Expected value of tolls per hour We expect 2x 60 = 120 motorcycles per hour, and each will pay £1. Doing the equivalent calculation for cars and goods vehicles, the expected value of tolls collected per hour is: 260 x £1+5 x 60 x £2 +1.5 x 60 x £5 = 60 x (2x £1+5 x £2+1.5 x £5) = £1170 (vi) Probability of more than 2 goods vehicles in a minute The number of goods vehicles arriving in a minute will follow a Poisson(1.5) distribution. So, the probability of more than 2 arriving in a minute is: P(N > 2) =1-{P(N = 0) +P(N = 1) + P(N = 2)} wt-feronsen# =1-3.625e"5 = 0.1912 (vii) Probability that exactly £4 is collected in a minute We can work out this probability by listing the events that will generate exactly £4 in tolls, together with their respective probabilities. If we write M, C and G for motorcycle, car and goods vehicle, these are: 4 4M,0C,0G > Zee xe xent5 =Rets = 0.0001356 2M,1C,0G 0M, 2C,0G The total probability for these is 0.0047137. © IFE: 2021 Examinations Page 31Subject CT4 April 2015 Question 1 () Simple random walk Asimple random walk is a process Xo, X;,Xp,... where Xo =0 and X_=Yy+Yp+-+Yq for n=1,23,.... The Y;'s are independent identically distributed random variables with distribution: _ {+1 with probability p ‘“\-1 with probability 1-p for some fixed probability 0

0 with the following properties: 1. No=0 2. _N, has independent increments 3. N; has Poisson distributed stationary increments where: © IFE: 2021 Examinations Page 45‘Compound Poisson process ‘A compound Poisson process X; is defined as: N, X= LY/.t20 ia where N,,t20 is a Poisson process and Y;, 21 is a sequence of ID random variables. Properties of specific processes. Time State ‘. domain space Markov | Stationary White Discrete or | Discrete or noise | continuous | continuous | _Y®S vee Simple random Discrete Discrete Yes No walk General random Discrete | Continuous Yes No walk Poisson . ‘ process Continuous | Discrete Yes No Compound - Poisson | Continuous sede Yes No process Processes of mixed type A process of mixed type is one that operates in continuous time but that can also change value at predetermined discrete instants. Counting processes Acounting process, X(t), is a stochastic process in discrete or continuous time, whose state space is the set of whole numbers {0, 1.2,...}, with the property that X(t) is a non-decreasing function of t. A Poisson process is an example of a counting process. Page 46 © IFE: 2021 Examinations