Do not turn over until you are told to do so by the Invigilator.

Transcription

1 UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination SETS, NUMBERS AND PROBABILITY MTHA4001Y Time allowed: 2 Hours Attempt QUESTIONS 1 and 2, and THREE other questions. Janacek statistics tables are available on your desk. Notes are not permitted in this examination. Do not turn over until you are told to do so by the Invigilator. MTHA4001Y Module Contact: Dr Davide Proment, MTH Copyright of the University of East Anglia Version: 1

2 (i) (a) Suppose that a, b, d, m, n are integers. Give the definition of what is meant by saying that d is a divisor of a. Using this, prove that if d is a divisor of a and d is a divisor of b, then d is a divisor of ma + nb. (b) Use the Euclidean algorithm to find the greatest common divisor d of 225 and 63. Hence (or otherwise) find integers m, n with 225m + 63n = d. (c) Are there integers s, t such that 225s + 63t = 17? carefully. Explain your answer [12 marks] (ii) Prove by induction that n 3 n is a multiple of 6 for all natural numbers n. [8 marks] MTHA4001Y Version: 1

3 (i) In the framework of modern probability, give the definition of two disjoint events and state Kolmogorov s three axioms of the modern theory of probability. (a) Use Kolmogorov s axioms to prove that P ( ) = 0. (b) Use Kolmogorov s axioms to prove that if A B then P (A) P (B). You may assume that P (C D) = P (C) + P (D) holds for any disjoint events C and D. [12 marks] (ii) At UEA Sportspark a survey showed that 25% of customers use the swimming pool and 75% go to the gym, while 90% do at least one of these activities. If a customer is selected at random, find the probability that they: (a) do both activities; (b) go to the gym and do not use the swimming pool; (c) use the swimming pool, given that they go to the gym; (d) do not go to the gym, given that they use the swimming pool. [8 marks] MTHA4001Y PLEASE TURN OVER Version: 1

4 (i) (a) State (but do not prove) the Fundamental Theorem of Arithmetic. (b) Prove that there are infinitely many prime numbers. (ii) Suppose that A is a non-empty set and is a relation on A. Define what it means for to be reflexive, symmetric and transitive. In each of the following cases, decide which (if any) of these properties the given relation has. Give reasons for your answers. (a) A = N and a b gcd(a, b) = 1 (for a, b N ). (b) A = C and a b a + b R (for a, b C ). 4. (i) Suppose that A, B are sets and f : A B is a function. Write down what it means for f to be surjective and what it means for f to be injective. For each of the following functions decide whether it is injective, surjective, both or neither. You should give brief reasons for your answers. (a) f : R R where f(x) = exp(x) for x R. { n/3 if 3 n, (b) g : Z Z where g(n) = n if 3 n. (ii) (a) State (but do not prove) Fermat s Little Theorem. (b) Compute the remainder when is divided by 31. (c) Find x Z such that 17x 1 (mod 77). MTHA4001Y Version: 1

5 (i) Let X be a Poisson random variable with parameter λ. The probability mass function for X is P (X = k) = λk e λ. k! (a) Show that P (X = k) = 1. k=0 (b) By assuming (a) is true, calculate E(X). [8 marks] (ii) Students travelling to the city arrive at the UEA bus stop according to a Poisson process of intensity 10 per 5 minutes between 17:00 and 19:00, and of intensity 3 per 10 minutes during the rest of the day. (a) What is the probability that at least 8 students arrive at the bus stop between 18:00 and 18:03? (b) What is the probability that at most 12 students arrive at the bus stop between 9:00 and 9:30? (c) Suppose that no students are the bus stop at 16:57. What is the probability that exactly 1 student arrives by 17:01? [12 marks] MTHA4001Y PLEASE TURN OVER Version: 1

6 (i) (a) Define expectation E(X) and variance V (X) of a continuous random variable X. A random variable X is said to have a normal N(µ, σ 2 ) distribution with mean µ and variance σ 2 if its probability density function is f(x) = 1 ] (x µ)2 exp [. 2π σ 2 σ 2 (b) Show that the probability density function f(x) satisfies the first Kolmogorov axiom of modern probability. (c) By rigorously evaluating the expectation E(X), prove that it is equal to the mean µ. You may use the result e s2 /2 ds = 2π. (ii) The standard normal random variable Z is a particular case of normal random variable having mean µ s = 0 and variance σ 2 s = 1. Its cumulative density function is defined by Φ(z) = P (Z z) = 1 2π z e z2 /2 dz and its values are computed numerically and tabulated in the statistical tables. (a) Give the definition of cumulative distribution function F (x) of the normal random variable X having f(x) as a probability density function. (b) Then, explain why the following relation holds ( ) x µ F (x) = P (X x) = Φ, that is z x µ. σ σ (c) Consider the normal random variable Y which has mean µ = 10 and variance σ 2 = 49. Find P (Y 0) and P (Y < 3 Y > 20). END OF PAPER MTHA4001Y Version: 1

7 MTHA4001Y Feedback on Main Series Examinations SNAP Question 1 This question was done well. Almost everyone was able to define a divisor; marks were lost here for vague answers. Marks were lost on the proof portion for not specifying that mr + ns (or similar) was an integer. Almost everyone was able to do the Euclidean algorithm correctly. Problems arose in part (c), where you are asked to prove that there are no such s, t Z such that 225s + 63t = 17; lots of your proofs relied on 17 not dividing m, n from the previous question. This was not enough. The induction question was rarely attempted; the base case was generally well done, but the inductive step proved confusing. Marks were lost here for incorrect statements of P (k), a missed step or two, or simply giving up half way through the proof. Question 2 Overall not very well done. Most of you did not stated correctly all the the three Kolmogorov s axioms, with mistakes mostly in the third axiom. More than a third of you wrote that two events A and B are disjoint when P (A B) = 0; this is indeed correct but it is a consequence of the true definition which simply says that two events A and B are disjoint when A B =. Marks were also lost for not explaining following a clear and linear logic why P ( ) = 0. Q2(ii) was overall sufficiently well done. Question 3 Part (i) of this question was done poorly. Many of you remembered the Fundamental Theorem of Arithmetic in part (a) was something to do with the factorisation of natural numbers into primes. Fewer remembered that this factorisation must be unique, and fewer than that also wrote that this was up to the order to multiplication. All three parts were required for full marks. Part (b) was not attempted in general; for those who did attempt it, the contradiction derived was usually a little sketchy. You should use the Fundamental Theorem of Arithmetic to derive your contradiction, as in the lecture notes. Part (ii) of this question was a lot better. For the most part, marks were lost in the definitions for lack of quantifiers (so a, b A). For the relations given, this was done well. Although marks were lost mainly for not providing numbers as counterexamples, this was a minority and I was very impressed with the correct use of counterexamples throughout the question by the majority of students. Question 4 In part (i), most of you were able to correctly write down the definitions of surjective and injective functions; this was a marked improvement on previous years, so well done! However, when it came to using these definitions, a significant amount of students were not able to prove that e x was injective; mostly due to skipped steps, vague proofs, or graphical arguments (which was mildly disappointing). The use of counterexamples was again a highlight of this question. There were almost no problems with part (ii); however, most of you forgot to write either a Z or p a prime in Fermat s Little Theorem in part (a). Part (b) was done well in general; marks were lost here for arithmetic errors. Similarly to the Euclidean Algorithm in question 1, part (c) was done well. Question 5 Q5(i) was overall well done but some marks were lost for not using the right notation (for instance in the summation) or not being clear enough in your explanations. One note: the first letter of the word MO: Dr Davide Proment 1/1 2017

8 MTHA4001Y Feedback on Main Series Examinations SNAP Maclaurin must be in capital letters as it comes from the Scottish mathematician Colin Maclaurin. About half of you attempted Q5(ii)(c) doing it right but marks were taken for not explaining that the probability is simply given by the sum of the two events E 1 = {1 student arrives in 16:57-17:00 and no students arrive in 17:00-17:01} and E 2 = {no students arrive in 16:57-17:00 and one student arrives in 17:00-17:01} because E 1 and E 2 are independent. Question 6 Not many of you attempted Q6: half of those did it very well, and the rest did it quite badly. Marks were lost in Q6(i)(b) for saying only that f(x) > 0 and not defining the generic event A that needs to satisfy P (A) 0. Marks were also lost for mistakes in the calculations of Q6(i)(c). MO: Dr Davide Proment 2/1 2017