Attempt QUESTIONS 1 and 2, and THREE other questions. penalised if you attempt additional questions.

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. penalised if you attempt additional questions. You will not be Janacek statistical 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) Prove by induction that for all natural numbers n, (3n 2) = n(3n 1). 2 [8 marks] (ii) (a) Suppose 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 321 and 75. Hence (or otherwise) find integers m, n with 321m + 75n = d. (c) Find integers s, t such that 321s + 75t = 15. [12 marks] MTHA4001Y Version: 1

3 (i) Let E be a random experiment with sample space S. Let A be any non-impossible event in S, and let E 1, E 2,..., E n be a partition of S. (a) Prove the Law of Total Probability, that is, prove that P (A) = n P (A E i )P (E i ). i=1 (b) State and prove Bayes Theorem. (ii) Three six-sided (standard) dice lie on a table. Two are fair dice; the other one is loaded so that it always rolls a number greater than two, with all possible outcomes being equally likely. (a) What is the probability of obtaining an odd number rolling the loaded die? (b) Among the three dice one is taken at random. What is the probability of rolling a three? (c) Among the three dice one is taken at random and a four is rolled. What is the probability that this is the loaded die? MTHA4001Y PLEASE TURN OVER Version: 1

4 (i) (a) State (but do not prove) the Fundamental Theorem of Arithmetic. (b) Prove that there is no rational number q with the property that q 2 = 3. Is there a rational number r with 3r 2 = 1? Explain your answer. (ii) Suppose A and B are sets and f : A B is a function. Define what is meant by f being surjective and what is meant by f being injective. For each of the following functions decide whether it is injective, surjective (or both, or neither). Give brief reasons for your answers. (a) f : R R where f(x) = cos(2x) for x R. (b) g : Z Z where g(x) = { x 1 x + 1 if x is even if x is odd. 4. (i) Suppose that A is a non-empty set and is a relation on A. Give the definitions of what is meant by saying that is 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 = Q and a b a b Z (for a, b Q). (b) A = Z and a b 7 4a + 3b (for a, b Z). (ii) (a) State (but do not prove) Fermat s Little Theorem. (b) Compute the remainder when is divided by 19. (c) Find an integer x Z such that 15x 1 (mod 32). MTHA4001Y Version: 1

5 (i) Let X be a binomial random variable based on n independent and identical trials with probability p of success in each. function P n (X = k) = p n (k) = Knowing that X has probability mass ( ) n p k (1 p) n k, k 0 k n and recalling the definitions and properties of expectation and variance, show that: (a) E(X) = np; (b) V (X) = np (1 p). [8 marks] (ii) Consider the random experiment of rolling two six-sided (standard) fair dice and adding the numbers obtained as a result. (a) Show that the probability of obtaining a number greater than 7 as a result is p = (b) The random experiment is repeated 5 times. What is the probability that the result is a number greater than 7 at most 3 times? (c) The random experiment is repeated 7 times. What is the probability that the result is a number greater than 7 at least 4 times? (d) The random experiment is repeated 4 times. What is the probability that the result is a number greater than 7 exactly 2 times? [12 marks] MTHA4001Y PLEASE TURN OVER Version: 1

6 (i) The continuous random variable T defining the time elapsing between two successive events in a Poisson process of intensity α has probability density function { α e α t, t 0 f(t) = 0, t < 0, with α > 0. (a) Prove that f(t) is a probability density function by showing that it satisfies the first two of Kolmogorov s axioms of modern probability. (b) Give the definition of expectation E(T ) of a continuous random variable T and compute it using the given probability mass function f(t). (c) Give the definition of cumulative distribution function F (t) for a continuous random variable T and show that, using the definition of f(t), it results in { 1 e α t, t 0 F (t) = 0, t < 0, with α > 0. [12 marks] (ii) Consider the particular case having α = Making use either of the probability distribution function f(t) or its cumulative distribution F (t), evaluate: (a) P (T 3) ; (b) P (T 5) ; (c) P (T 8 T > 5). [8 marks] END OF PAPER MTHA4001Y Version: 1

7 MTHA4001Y Feedback on Main Series Examinations SNAP Question 1 Overall students did well on this question. In part (i) marks were lost for not setting out the induction argument in the correct way. You need to assume P (k), for some k, and then deduce P (k + 1). Some marks were lost in (ii)(c) since some students accidentally multiplied the equation from the previous part by 3, rather than by 5. Question 2 Overall not well done, especially Q2(i). Most of the students were probably expecting to be asked to state Kolmogorov s axioms, instead stating and proving the Law of Total Probability and Bayes s theorem. This is, in my opinion (Dr Davide Proment) a clear evidence that the students prepare the exam based on the previous years scripts (which did not contained the Law of Total Probability and Bayes s theorem) instead of revising on the lecture notes and their examples. Q2(ii)(a-b) were sufficiently well done, but marks were lost for not explaining how the final results were obtained. Q2(ii)(c) was probably correctly answered by only half of the students. In order to fully address Q2 one should have only revised the Law of Total Probability and Bayes s theorem, as Q2(ii) were a straightforward applications to them. Question 3 Some marks were lost in (i)(a) for forgetting to mention that the Fundamental Theorem of Arithmetic says that each natural number can be written as a product of primes in a unique way (up to the order of multiplication). For the proof in (i)(b) marks were lost for not mentioning that you are using the fact that 3 is a prime number to deduce that 3 a 2 implies 3 a. The last part of (i)(b) was the part that students struggled with the most. The key observation for this part is that if r is a non-zero rational number then 1/r is also a non-zero rational number. In part (ii) some students were not able to state correctly the definitions of injective and surjective function. It is important to know basic definitions like this for the exam. As happens almost every year at least one student said that injectivity means If a = b then f(a) = f(b). It should hopefully be clear to anyone that every function has this property, and it is certainly not the correct definition of injective function. Most students did very well with (ii)(a) but found (ii)(b) more difficult. One of the best answers to (ii)(b) was to prove the function is a bijection by finding an inverse for it. Question 4 Marks were lost in (i) for not correctly stating the definitions of reflexive, symmetric and transitive. Typical errors included forgetting to say for all a A when defining reflexivity; or other logical issues like writing a b and b a instead of if a b then b a in the definition of symmetry. Regarding the examples, (i)(a) was dealt with well in general, while students struggled more with (i)(b). A very common error was that some students claimed that If 7 4a + 3b then either a = b or 7 a and 7 b. But this is not true. For example if we set a = 8 and b = 6 then 4a+3b = = 14 which is divisible by 7, while a b and neither of them is divisible by 7. Part (ii) was done very well by those students who attempted it. MO: Dr Davide Proment 1/

8 MTHA4001Y Feedback on Main Series Examinations SNAP Question 5 Q5(i) was not well done at all: probably less than half of the students that attempted this part wrote the correct formula for the expectation. Marks were lost in not explaining why the expectation and variance of a Bernoulli trial random variable are p and p (1 p) respectively. Q5(ii) was overall well done. Question 6 Not many students attempted Q6 but half of those did it quite well. Some marks were lost in Q6(i)(a) for saying only that f(t) > 0 and not defining the generic event A that needs to satisfy P (A) 0; this is a pity as the previous year s exam feedback highlighted specifically this point! Marks were also lost for mistakes in calculations of Q6(ii), especially part (c). MO: Dr Davide Proment 2/