MATH S350 TMA 01 Hints Question 1 Let A and B be the event that water is contaminated with excessive amount of pollutant A and pollutatnt B respectively. Remarks: 1. Half of those communities whose drinking water containing excessive amount of pollutant A will also contain excessive amount of pollutant B which means that P(B|A)=0.5. 2. Since these two pollutants in drinking water are not independent which means that P(A and B) P(A)P(B) 3. For (a)(i), you only need to concern that the water is contaminated by excessive amount of pollutant B, ignore to consider the pollutant A. (a)(i) (a)(ii) (b)(i) (b)(ii) (b)(iii) (b)(iv) (c) P(selected person will have health problem | B) P(A | B) P(A and B) P(A and Not B) P(Not A and B) P(Not A and Not B) P(selected person will have health problem) = P(selected person will have health problem | A and B) P(A and B) + ? + ? Question 2 Remarks: 1. Since X 1 , X 2 , ... be independent and identically distributed random variables with an exponential distribution with parameter , S n ~ ( n, ) . (a) P( N (t ) n) P( N (t ) n) P( N (t ) n 1) P( S n t ) P( S n 1 t ) (b) P(t u S k t , t S k 1 t v) = P(t u S k t , t S k X k 1 t v ) ( S k 1 X 1 ... X k X k 1 Sk X k 1 ) t t v x = u x t t f S k ( x) f X k 1 ( y ) dydx where f S k (x ) and f X k 1 ( y ) are p.d.f of (k , ) and M ( ) respectively. (c) P (Vt z1 , Z 2 z2 ,..., Z m zm ) = P( N (t ) k , Vt z1 , Z 2 z2 ,..., Z m zm ) k 0 Page 1 = P( N (t ) k , Sk 1 t z1 , X k 2 z2 ,..., X k m zm ) k 0 ( Vt S N (t ) 1 t and Z i X N (t ) i for i = 2, 3, ...) = P( S k t , t Sk 1 t z1 , X k 2 z2 ,..., X k m zm ) k 0 ( t ) k t e c.d.f. of Sk 1 k! k 0 = c.d.f of X k 2 z2 ,..., X k m zm (d) Given that U t and Vt are independent, you are asked to show that P( U t u )=?. P (U t u , Vt v ) = P( N (t ) k , S k t u , S k 1 t v) k 0 = P(t u S k t , t S k 1 t v) = Ans. in (b) =... Question 3 Remarks: 1.The moment generating function of a random variable X, where it exists, is given by M X (t ) E (etX ) , The moment generating functions of a discrete r.v. and a continuous r.v. are respectively M X (t ) etx f ( x) , x M X (t ) e tx f ( x ) dx It is often abbreviated as mgf. The subscript X indicates it is the moment generating function of the random variable X and can be dropped 2. Given a sequence of n random variables X 1 , X 2 , ... X n and their joint probability distribution or density, we shall find the probability distribution or the density of some random variable Y = u ( X 1 , X 2 , ... X n ). In common practice, three methods (namely, the distribution function technique, the transformation technique and the moment generating function technique) will be investigated. Moment generating function technique which is very useful to deal with sums of independent random variables, but one possible shortfall is that one has to identify the moment generating function in order to obtain the distribution of Y Page 2 (a) tZ E (e ) e zt 1 e 2 z2 2 dz = ... ? 1 Consider: X Z , change Z as subject and sub. the result to the given p.d.f. of the random variable Z, you will get the p.d.f. of random variable X. The m.g.f. of random variable X is E (etX ) = ?. The m.g.f. of random variable X is E (e X ) = ?. (b) FY ( y ) P(Y y ) P(e X y ) P( X ln y ) FX (ln y ) (distribution function technique) The c.d.f. of Y is FY ( y ) FX (ln y ) Differentiate both sides w.r.t. y, you will obtain the p.d.f. of y. E (Y ) yfY ( y ) dy 0 (c) p.d.f. of T: FT (t ) P(T t ) P ( Z 2 t ) P ( t Z t ) =? m.g.f. of T = ? Question 4 (a) (b)(i) (b)(ii) (b)(iii) (b)(iv) (b)(v) Tutorial Note 2 page 8 Tutorial Note 2 example 6(a) Tutorial Note 2 example 6(c), (d) Tutorial Note 2 example 6(c), (d) Tutorial Note 2 page 10 Tutorial Note 2 page 10 Question 5 (a)(i) (a)(ii) (b) (c) Handbook page 18 point 15 n 0 1 2 P(N=n) 0.0672 0.1815 0.245 F(n) ? ? ? Since u = 0.37336, the number of diseased trees in this region is ?. Tutorial Note 2 example 8 Tutorial Note 2 example 9 Page 3 Page 4 MATH S350 Tutorial 2 Note Book 2: Modelling events in time and space 1. The Bernoulli Process A Bernoulli process is a sequence of Bernoulli trials in which trials are independent; the probability of success remains the same from trial to trial. Remarks: 1. A Bernoulli process has the memoryless property ( i.e. the occurrence of events in the future does not depend in any way on how many events have occurred in the past or on when they occurred.) 2. Examples of Bernoulli processes include: a sequence of rolls of a die where success is a 'six'; testing items off a production line where success is a 'good' item. 2. The Poisson Process A poisson process is a model for the occurrence of events in continuous time in which the following assumptions are made: Events occur singly; The rate of occurrence of events remains constant; The incidence of future events is independent of the past. Remarks: For a Poisson process, the number of events that occur during a time interval of length t has a Poisson Distribution, and the waiting time between successive events has a Exponential Distribution. Page 1 MATH S350 Tutorial 2 Note Remarks: Properties of Exponential Distributions Example 1 At the 'Hot Drink' counter in a cafeteria both tea and coffee are sold. The number of cups of coffee sold per minute may be assume to have a Poisson distribution with mean 2 and the number of tea sold per minute may be assume to be an independent Poisson distribution with mean 1.5. (i) Calculate the probability that in a given one-minute period (1) exactly l cup of coffee and l cup of tea are sold. (2) exactly 3 cups of hot drinks are sold. Solution (1) The required probability = 2e 2 1.5e 1.5 = 0.0906 (2) The required probability = 1.50 e 1.5 23 e 2 1.51 e 1.5 2 2 e 2 1.52 e 1.5 21 e 2 1.53 e 1.5 20 e 2 = 0.2158 0! 3! 1! 2! 2! 1! 3! 0! (ii) Suppose that in a given one-minute period exactly 3 cups of hot drinks are sold. Calculate the probability that (1) these are all cups of coffee. (2) 1 cup is coffee and 2 cups is tea. Solution 23 e 2 3! = 0.1866 0.2158 e 1.5 (1) The required probability = 1.52 e 1.5 2e 2! (2) The required probability = = 0.3149 0.2158 2 Page 2 MATH S350 Tutorial 2 Note Example 2 Events occur according to a Poisson process in time with, on average, sixteen minutes between events. (a) Write down the probability distribution of the waiting time T between events, where T is measured in hours. (b) Write down the probability distribution of the number of events in any hour-long interval. (c) If there were seven events between 2pm and 3pm yesterday, calculate the probability that there was at most one event between 3pm and 4pm. (d) Calculate the probability that the waiting time between successive events will exceed half an hour. Solution Page 3 MATH S350 Tutorial 2 Note Example 3 Data collected on major volcanic eruptions in the northern hemisphere give a mean time between eruptions of 29 months. Assume that such eruptions occur as a Poisson process in time. (a) Calculate the expected number of eruptions during a five-year period. (b) Calculate the probability that there are exactly two eruptions during a five-year period. (c) Calculate the probability that at least three years pass after you have read this exercise before the next eruption. Solution 3. The multivariate Poisson process A multivariate Poisson process is a Poisson process in which each event may be just one of several different types of event. For instance, in traffic research, an event may correspond to the passage of a vehicle past a point of observation by the side of a road. The vehicles may themselves be classified as private cars, lorries, buses, motorcycles, and so on. Page 4 MATH S350 Tutorial 2 Note Example 4 Customers arrive at a small post office according to a Poisson process at the rate of eight customers an hour. In general, 70% of customers post letters, 5% post parcels and the remaining 25% make purchases unrelated to the postal service. (a) At what rate do customers arrive at the post office to post parcels? Solution (b) Calculate the probability that the interval between successive customers arriving to post parcels is greater than an hour. Solution (c) Calculate the probability that over a three-hour period, fewer than five customers arrive to post letters. Solution Page 5 MATH S350 Tutorial 2 Note (d) Calculate the median waiting time between customers arriving to post something (either a letter or a parcel). Solution Example 5 A university tutor has noticed that over the course of an evening, telephone calls arrive from students according to a Poisson process at the rate of one every 90 minutes. Independently, calls arrive from members of her family according to a Poisson process at the rate of one every three hours, and calls from friends arrive according to a Poisson process at the rate of one call per hour. She does not receive any other calls. (a) Calculate the probability that between 7 pm and 9 pm tomorrow evening, the tutor's telephone will not ring. Solution Page 6 MATH S350 Tutorial 2 Note (b) Calculate the probability that the first call after 9pm is from a student. Solution (c) Given that she receives four telephone calls one evening, calculate the probability that exactly two of the calls are from members of her family. Solution 4. The non-homogeneous Poisson process In probability theory, an inhomogeneous Poisson process (or non-homogeneous Poisson process) is a Poisson process with rate parameter (t) such that the rate parameter of the process is a function of time. Page 7 MATH S350 Tutorial 2 Note Example: Mining accidents This graph is roughly linear for the first 14 000 days (up to about 1890), but the gradient then becomes less steep. This means that the rate of occurrence of accidents decreased. This is an example of a process where the rate of occurrence of events is not constant, but changes with time. Example 6 Suppose that the accident rate at time t of a young girl learning to ride a bicycle is given by: (t) = 24/(2 + t) , t 0, where t is measured in days. (a) Find the expected number of accidents in the first t days. Solution Page 8 MATH S350 Tutorial 2 Note (b) Calculate the expected number of accidents during the first week. Solution (c) Calculate the expected number of accidents during the third week. What is the probability that the girl has eight accidents in the third week? Solution (d) Calculate the probability that the fourth week is free of accidents. Solution Page 9 MATH S350 Tutorial 2 Note 5. Simulation for a non-homogeneous Poisson process Page 10 MATH S350 Tutorial 2 Note 6. The compound Poisson process Example: Visitors to a restaurant Between 12 noon and 2 pm, private cars call at a roadside restaurant according to a Poisson process at the rate of one car every four minutes. If X(2) is the number of cars that arrive in the two-hour period, then X(2) has a Poisson distribution with parameter 30. Suppose that the number of occupants in each car is an observation on the discrete random variable Y with the geometric distribution G1(0.4). Then S(2), the total number of visitors to the restaurant over the two-hour period, is given by S(2) = Y1 +Y2 + X(2), +Y where X(2) ~ Poisson(30) and each Yi has the given geometric distribution. Example 7 According to the visitors to a restaurant example, Yi, the number of occupants in each car calling at a roadside restaurant, is an observation on a geometric random variable: Yi ~ G1(0.4). Therefore Page 11 MATH S350 Tutorial 2 Note 7. Patterns in space The main way in which these patterns differ from patterns of events in time is that they usually occur in more than one dimension. Most of the patterns discussed in this book are two-dimensional patterns spread over an area of land - for example, animals grazing in a field, oak trees in a wood, or shops in a town. Example 8 The locations of plants in a rectangular area measuring 6 metres by 5 metres are recorded in an investigation into their reproductive behaviour. The numbers of plants in square metre quadrats are as follows. 7 7 7 5 4 5 3 3 2 9 5 6 2 4 4 4 3 4 4 5 7 7 5 11 3 4 1 4 1 8 Use these data to investigate whether the plants can reasonably be supposed to be randomly located over the rectangle. Use a 5% significance level. Solution Page 12 MATH S350 Tutorial 2 Note Example 9 The locations of plants in a rectangular area measuring 6 metres by 5 metres are recorded in an investigation into their reproductive behaviour. Twenty of the plants are selected at random, and the distance from each to its nearest neighbour is recorded. The 20 distances (in metres) are as follows. 0.20 0.27 0.31 0.32 0.37 0.07 0.23 0.24 0.16 0.33 0.35 0.45 0.34 0.30 0.36 0.16 0.23 0.57 0.35 0.33 At the same time, 20 points are selected at random in the region, and the distance from each to the nearest plant is recorded. These 20 distances (in metres) are as follows. 0.10 0.16 0.36 0.21 0.36 0.32 0.15 0.29 0.07 0.26 0.14 0.32 0.33 0.10 0.30 0.26 0.32 0.36 0.23 0.16 Use these data to investigate whether the plants can reasonably be supposed to be randomly located over the rectangle. Use a 5% significance level. Solution Page 13 MATH S350 April 2014 Presentation TMA 01 Cut-off Date: 16 Oct 2015 This assignment covers Book 1 to Book 2. Question 1 (Book 1) - 20 marks Drinking water may be contaminated by two pollutants. In a given community, the probability of its drinking water containing excessive amount of pollutant A is 0.1 whereas that of pollutant B is 0.2. When pollutant A is excessive, it will definitely cause health problem; however, when pollutant B is excessive, it will cause health problem in only 20% of the population who has low natural resistance to that pollutant. Also, data from many similar communities reveal that the presences of these two pollutants in drinking water are not independent; half of those communities whose drinking water containing excessive amount of pollutant A will also contain excessive amount of pollutant B. (a) Given that the water is contaminated by excessive amount of pollutant B, (i) Find the probability that a randomly selected person from the community will have a health problem. [2] (ii) Find the probability that water will also be contaminated with excessive amount of pollutant A. [4] (b) Find the following probabilities: (i) Water is contaminated with excessive amount of both pollutant A and B. (ii) Water is contaminated with excessive amount of pollutant A but not B. (iii) Water is contaminated with excessive amount of pollutant B but not A. (iv) Water is neither contaminated with excessive amount of pollutant A nor B. [2] [3] [3] [2] (c) Suppose a resident is selected at random from this community, what is the probability that he/she will suffer health problem from drinking water? You may assume that a person's resistance to pollutant B is innate, which is independent of the event of having excessive pollutant in the drinking water. [4] Be sure to define the events involve and to state clearly the results from the materials that you use. Question 2 (Book 1) - 20 marks This question is concerned with another definition of a homogeneous Poisson process. We starts with exponentially distributed random variables and we can show that the number of events occurrence in a fixed time interval has a Poisson distribution. The later part of this question proves the interesting waiting time paradox. A motivating example is that when we go to a bus stop and do not see a bus coming, what is the distribution of the time we need to wait for the next bus? Let 1 , 2 , ... be independent and identically distributed random variables with an exponential distribution with parameter > 0. Let 0 = 0, and for n = 1,2, ..., let = =1 Define, N(0) = 0 and, for all t > 0, N() = max{: }. (a) Show that P(() = ) = (You may use the fact that 0 () 1 (1)! () ! = () ! + 0 () ! ) [5] (b) For all integer k, t > 0, u (0, ] and v > 0, show that P( < , < +1 () (( )) + ) = (1 ). ! [5] (c) Let = ()+1 and let for all i = 2,3, ..., = ()+ . Then for all m, the random variables , 2 , ... , are independent and identically distributed with an exponential distribution with parameter . (Hint: Show that P( 1 , 2 2 , ... , ) = =1(1 )) [5] (d) Define = () . Show that the distribution of is given by for 0 < P( ) = {1 1 for Moreover, and are independent. [5] Question 3 (Book 1) - 15 marks The continuous real valued random variable Z has pdf given by () = 1 2 exp { 2 } , . 2 (a) For a small t > 0, find the moment generating function (mgf), E( ), of , and also the pdf and mgf of the random variable , where = + for parameters and > 0. Hence deduce the expectation of (), where () = . [4] [2] (b) Suppose that is a random variable defined in terms of by = . Find the pdf of and show that the expectation of is exp { + 1 }. 22 [5] 2 (c) Let be a random variable defined by = . Find the pdf and mgf of . [4] Question 4 (Book 2) - 25 marks (a) The arrival of cars at a motorway service area to buy fuel during a twenty-four hour period may be modelled by a non-homogeneous Poisson process with rate (). Draw a sketch showing what you think the arrival rate () might look like from midnight to midnight on a weekday, and justify your sketch briefly. [3] (b) The arrival of cars to buy fuel at a small all-night gauge between midnight and 5am on a weekday may be reasonably modelled as a non-homogeneous Poisson process with rate () = 3 100+3 , where is the time in minutes after midnight. (i) Show that the expected number of cars arriving by minutes after midnight is ( ) = log(1 + 0.03). [4] (ii) (iii) Find the probability that one car arrives between midnight and 12:30am. Find the probability that at least two cars arrive between 4am and 5am. (iv) Show that the simulated times after midnight 1 , 2 , ..., at which cars arrive may be obtained from the recurrence relation +1 = (v) 3 +100 3(1) [3] [4] , where is a random observation from the uniform distribution (0,1). [5] Simulate the times of arrival of cars between midnight and 1am, using random numbers 0.0406, 0.46030, 0.23751 and 0.61380, give the times to the nearest minute. [6] Question 5 (Book 2) - 20 marks (a) In a particular wood, diseased trees may be assumed to be randomly located according to a two-dimensional Poisson process with density 18 per square kilometer. A rectangular region of the wood 500 meters long by 300 meters wide is marked out for detailed study. (i) (ii) What is the probability distribution of the number of diseased trees in this region? [2] Simulate the number of diseased trees in this region using the number = 0.37336 which is a random observation from the uniform distribution (0,1). [2] (b) The wood is split into 24 equal-sized quadrats and the number of trees of a particular species in each quadrat is counted. The results are given below: 6 9 5 11 7 10 9 7 8 7 4 8 7 7 12 9 6 6 8 6 8 5 9 7 Commenting on the suitability of whatever test you adopt, use these data to investigate whether the trees could reasonably be supposed to be randomly positioned in the wood. If your analysis suggests that the trees are not randomly located, then say how you think they are located. Use a 5% significance level and state clearly the critical region for your test. [8] (c) In an investigation into the disposition of a certain species of small shrub in the wood, ten R-distances (point-to-nearest-object distances) and ten S-distances (object-to-nearest-object distances) are calculated. These are shown below: R-distances S-distances 10.7 7.5 9.9 12.0 19.9 18.7 20.3 19.7 13.1 10.2 6.1 9.7 11.4 14.3 13.6 15.5 17.1 18.8 14.7 16.8 Use these data to investigate whether shrubs of this species can be reasonably supposed to be randomly located over the area of investigation. If your investigation suggests that they are not, then say how you think the shrubs are located. Use a 5% significance level and state clearly the critical region of your test. [8]