EEI Page 5 of 10 SECTION B [Note: Statistical Tables A1 and A2 are provided at the end of this question] B1. Answer all parts of this question (a) Let A, B and C be three arbitrary events. Write down the missing term represented by the x in the following: (i) Only A occurs: (2 marks) (ii) At least one event occurs: (2 marks) (iii) At least two events occur: ( (2 marks) (iv) One and only one event occurs: ( (2 marks) (v) Not more than two events occur: (2 marks) (b) There is a plan to build a bridge across the Messina Strait. Consideration, however, must be given to the threat that earthquakes pose to the bridge. It is thought that the annual number of earthquakes of magnitude greater than 5 may be modelled by a Poisson distribution with rate λ > 0; that is letting X be the random variable that represents the number of earthquakes in a period of time t years, , x = 0, 1, 2, …. For X ~ Poisson (λt), E(X) = var(X) = λt. The number of major earthquakes (magnitude > 5) occurring in each ten year period from 1701-1980 is given in Table B1.1 below: Table B1.1 Number of major earthquakes 0 1 2 3 Frequency 13 11 2 2 ...cont./ EEI Page 6 of 10 (i) Using the method of moments, estimate λ. (2 marks) (ii) Assuming the rate λ will remain constant over time, derive an expression for the probability of observing at most one earthquake between 2015 and 2035. Using the method of moments estimate for λ, calculate this probability. (4 marks) (iii) It is known that if the number of events occurring in a period of length 1 unit follows a Poisson (λ) distribution, the inter-arrival times between successive events Y have an Exponential distribution with probability density function f(y) = λe-λy, y > 0, λ > 0. Derive an expression for the probability that the time between successive earthquakes of magnitude 5 is greater than 15 years. Calculate this probability, using the method of moments estimate for λ from part (i). (4 marks) EEI Page 7 of 10 B2. Answer all parts of this question Let X be the random variable that denotes the hourly wind speed (ms-1) at a measuring station. The cumulative distribution function of X is: (a) Show that X has the probability density function: (2 marks) (b) The mean of X is E(X) . The median of X is the value of x such at F(x; θ) = 1/2. Are the mean and median equal? Give reasons for your answer. (2 marks) (c) The likelihood function for n independent random variables X1, …., Xn, each with the pdf f(x;θ ) is: (θ) = f(x1;θ) x f(x2:θ) x …. x f(xn-1 θ) x f(xn;θ ) Explain the idea of maximum likelihood estimation. Why can the logarithm of the likelihood function be used to obtain a maximum likelihood estimate of θ? (3 marks) (d) Show that the maximum likelihood estimate of θ is = State how you would check this is a maximum, but you do NOT need to carry out the calculations to check this. (7 marks) (e) A random sample of ten wind speeds was observed: 20.5, 4.3, 6.0, 10.6, 6.3, 12.7, 11.2, 13.1, 12.2, 24.8 Calculate the maximum likelihood estimate of θ for this sample (2 marks) (f) Derive an expression for P(X < 5="" ǀ="" x="">< 10)="" in="" terms="" of="" θ.="" use="" the="" value="" of="" calculated="" in="" part="" (d)="" to="" estimate="" p(x="">< 5="" ǀ="" x="">< 10). (4 marks) eei page 8 of 10 b3. answer all parts of this question (a) suppose that x1 ~ n(1,2) and x2 ~ n(0,2) and that x1 and x2 are independent, and y = 2x1 - 2x2. (i) show that y ~ n(2, 16); (3 marks) (ii) calculate p(y ≤ 1); (3 marks) (iii) calculate p((y ≤2) (y≥ 3)). (4 marks) (b) the volumes of water (l) in a random sample of n = 7 drinking water containers were: 97.3, 98.1, 97.2, 96.7, 99.6, 98.0, 101.7 (i) assuming the volumes are normally-distributed, calculate a 95% confidence interval for the true mean volume. (7 marks) (ii) how would you respond to a claim that the true mean volume is equal to 100l? (3 marks) 10).="" (4="" marks)="" eei="" page="" 8="" of="" 10="" b3.="" answer="" all="" parts="" of="" this="" question="" (a)="" suppose="" that="" x1="" ~="" n(1,2)="" and="" x2="" ~="" n(0,2)="" and="" that="" x1="" and="" x2="" are="" independent,="" and="" y="2X1" -="" 2x2.="" (i)="" show="" that="" y="" ~="" n(2,="" 16);="" (3="" marks)="" (ii)="" calculate="" p(y="" ≤="" 1);="" (3="" marks)="" (iii)="" calculate="" p((y="" ≤2)="" (y≥="" 3)).="" (4="" marks)="" (b)="" the="" volumes="" of="" water="" (l)="" in="" a="" random="" sample="" of="" n="7" drinking="" water="" containers="" were:="" 97.3,="" 98.1,="" 97.2,="" 96.7,="" 99.6,="" 98.0,="" 101.7="" (i)="" assuming="" the="" volumes="" are="" normally-distributed,="" calculate="" a="" 95%="" confidence="" interval="" for="" the="" true="" mean="" volume.="" (7="" marks)="" (ii)="" how="" would="" you="" respond="" to="" a="" claim="" that="" the="" true="" mean="" volume="" is="" equal="" to="" 100l?="" (3="">