MTH4107 / MTH4207: Introduction to Probability

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1 Main Examination period 2018 MTH4107 / MTH4207: Introduction to Probability Duration: 2 hours Student number Desk number Make and model of calculator used Apart from this page, you are not permitted to read the contents of this question paper until instructed to do so by an invigilator. Write your solutions in the spaces provided in this exam paper. If you need more paper, ask an invigilator for an additional booklet and attach it to this paper at the end of the exam. You should attempt ALL questions. Marks available are shown next to the questions. Calculators are permitted in this examination. Please state on your answer book the name and type of machine used. Complete all rough work in the answer book and cross through any work that is not to be assessed. This is an OPEN BOOK exam permitted: any printed material, e.g. books any handwritten notes photocopies of any kind prohibited: using communication devices, e.g. laptops or mobile phones sharing material with other students Possession of unauthorised material at any time when under examination conditions is an assessment offence and can lead to expulsion from QMUL. If you are found to have hidden unauthorised material elsewhere, including toilets and cloakrooms, it shall be treated as an assessment offence. A mobile phone that causes a disruption in the exam is also an assessment offence. Exam papers must not be removed from the examination room. Examiners: W. Just Turn Over

2 Page 2 MTH4107 / MTH4207 (2018) This page is for marking purposes only. Do not write on it. Question Mark Comments 1 / 25 2 / 25 3 / 25 4 / 25 Total

3 MTH4107 / MTH4207 (2018) Page 3 Question 1. [25 marks] You have a fair and a biased coin. You toss the fair coin once. If it shows head you toss the fair coin again one time. If it shows tail you toss the biased coin one time. (a) Write down the sample space for this experiment. [5] {hh, ht,th,tt } (set 1, elements as pairs 2, complete set 2, but no special symbol for biased coin needed) Turn Over

4 Page 4 MTH4107 / MTH4207 (2018) (b) State the following events as subsets of the sample space: [6] A: Both tosses produce the same result B: I toss the same coin twice C: I toss two different coins A = {hh,tt } 2 B = {hh,ht} 2 C = {th,tt } 2

5 MTH4107 / MTH4207 (2018) Page 5 (c) The biased coin has probability 2/3 of showing head. Compute the probabilities of the events stated in part (b). [6] Using independence P(A) = = P(B) = = P(C) = = Turn Over

6 Page 6 MTH4107 / MTH4207 (2018) (d) Compute the conditional probability of the event B given the event A. [4] P(A B) = P({hh}) = P(B A) = P(A B) P(A) = 1/4 5/12 = 3 5 2

7 MTH4107 / MTH4207 (2018) Page 7 (e) Are the events B and C independent? Give a reason for your answer. [4] B C = /0 (disjoint events) thus P(B C) = 0 2 P(B C) = = P(B)P(C), hence events dependent. 2 Turn Over

8 Page 8 MTH4107 / MTH4207 (2018) Question 2. [25 marks] A bag contains 2 red balls, 3 red cubes, 3 blue balls, and 2 blue cubes. You pick 3 objects at random without replacement. (a) Compute the probability that you draw 2 blue balls. Simplify the result and state the result as a fraction. [6] Unordered sampling without replacement, sample space size ( 10) 3 = ( 3 ) ( 2 = 3 possibilities to sample 2 blue balls from 3 blue balls 1 and 7 ) 1 = 7 possibilities to sample a non blue ball from 7 objects 1 P(2 blue balls) = =

9 MTH4107 / MTH4207 (2018) Page 9 (b) Compute the probability that all the objects you draw are blue. Simplify the result and state the result as a fraction. [6] Unordered sampling without replacement, size of sample space ( 10) 3 = ) = 10 possibilities to sample 3 blue objects from 5 blue objects 2 ( 5 3 P(all blue) = = Turn Over

10 Page 10 MTH4107 / MTH4207 (2018) (c) Denote by C the random variable counting the number of cubes you draw. Compute the probability mass function of C. [4] P(C = 0) = P(C = 1) = P(C = 2) = P(C = 3) = ( 5 3) ( 10 ) 1 = ( 5 )( 5 1 ( 2) 10 ) 1 = ( 5 )( 5 2 ( 1) 10 ) 1 = ( 5 3) ( 10 3 ) 1 = 1 12

11 MTH4107 / MTH4207 (2018) Page 11 (d) Denote by R the random variable counting the number of red objects you draw. Compute the joint probability that R = 0 and C = 3. [4] R = 0 and C = 3 means all three objects are blue and all three objects are cubes 1. This event is the emptyset, as there are only two blue cubes 1 P(R = 0,C = 3) = 0 2 Turn Over

12 Page 12 MTH4107 / MTH4207 (2018) (e) State with reason whether the random variables R and C are independent. [5] Hence dependent random variables 1. P(R = 0) = P(all blue) = P(C = 3) = P(R = 0,C = 3) = = P(R = 0)P(C = 3) 2 12

13 MTH4107 / MTH4207 (2018) Page 13 Question 3. [25 marks] A bus operator operates two lines each connecting the city Xinji with Yiwu. Line 1 runs a bus A from Xinji to Ulanqab and a bus B from Ulanqab to Yiwu, so that passengers have to change at Ulanqab. Line 2 runs a bus C from Xinji to Weihui and a bus D from Weihui to Yiwu, so that passengers have to change at Weihui. Buses are running with probabilities P(A) = 0.9, P(B) = 0.8, P(C) = 0.7, P(D) = 0.8. Furthermore the company ensures that at least three of the four buses are running. (a) Using the inclusion-exclusion principle, or otherwise, compute the probability that I can travel via line 1 from Xinji to Yiwu. [5] P(A B) = 1 2 (since at most one bus is nonoperational) P(A B) 1 = P(A) + P(B) P(A B) 1 = = Turn Over

14 Page 14 MTH4107 / MTH4207 (2018) (b) Using the inclusion-exclusion principle, or otherwise, compute the probability that I can travel via line 2 from Xinji to Yiwu. [5] P(C D) = 1 2 (since at most one bus is nonoperational) P(C D) 1 = P(C) + P(D) P(C D) 1 = = 0.5 1

15 MTH4107 / MTH4207 (2018) Page 15 (c) Assuming that I take bus A. Compute the probability that I can take bus B when changing at Ulanqab. [5] P(B A) 2 = P(A B) P(A) 2 = = Turn Over

16 Page 16 MTH4107 / MTH4207 (2018) (d) Assume that I take bus C. Compute the probability that I can take bus D when changing at Weihui. [5] P(D C) 2 = P(C D) P(C) 2 = = 5 7 1

17 MTH4107 / MTH4207 (2018) Page 17 (e) Assume both buses, A and C are available at station Xinji. Which bus should I take to have a higher probability to reach the destination Yiwu. Justify your choice by comparing the relevant probabilities. [5] P(A C) = P(A) + P(C) 1 and P(A B C) = 1 (see (a) and (b)), hence P(B A C) = P(A B C) P(A) P(B) P(C) + P(A B) + P(A C) + P(B C) = P(A) + P(B) + P(C) 2 = Accordingly Hence and P(D A C) = P(A) + P(D) + P(C) 2 = P(B A C) = P(D A C) = P(B A C) P(A C) P(D A C) P(A C) = = = = are the same 1. Turn Over

18 Page 18 MTH4107 / MTH4207 (2018) Question 4. [25 marks] Let X be a random variable with X Geom(1/3). Let Y be another random variable which has Binomial(N,1/4) distribution where N is the value taken by the random variable X. (a) State the expectation value of the random variable X. [4] With p = 1/3 E(X) = 1 p 2 = 3 2

19 MTH4107 / MTH4207 (2018) Page 19 (b) Assuming X takes the value X = N compute the conditional expectation value of the random variable Y. [6] With p = 1/4 E(Y X = N) 2 = N p 2 = N 4 2 Turn Over

20 Page 20 MTH4107 / MTH4207 (2018) (c) Using the law of total probability, or otherwise, compute the expectation value of Y. [6] E(Y ) = = 1 4 N=1 = E(Y X = N)P(X = N) 2 NP(X = N) 1 = 1 N=1 4 E(X) 2

21 MTH4107 / MTH4207 (2018) Page 21 (d) Using the law of total probability, or otherwise, compute the expectation value of the product XY and hence compute the covariance of the random variables X and Y. [5] E(XY ) = = 1 4 N=1 NE(Y X = N) 1 P(X = N) N 2 P(X = N) N=1 = 1 4 E(X 2 ) 1 = 1 ( Var(X) + (E(X)) 2 ) 1 4 = 1 ( 2/3 4 1/9 + 1 ) = 15 1/9 4 1 Cov(X,Y ) = E(XY ) E(X)E(Y ) 1 = = 3 2 Turn Over

22 Page 22 MTH4107 / MTH4207 (2018) (e) Are the random variables independent? Give a reason. [4] Random variables are dependent 2 since covariance does not vanish 2.

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MTH6134 / MTH6134P: Statistical Modelling II

Main Examination period 2017 MTH6134 / MTH6134P: Statistical Modelling II Duration: 2 hours Student number Desk number Name and type of calculator used Apart from this page, you are not permitted to read