Avd. Matematisk statistik

EXAM IN SF1901 PROBABILITY THEORY AND STATISTICS, MONDAY AUGUST 14, 2017 AT 08.00–13.00.

Examiner: Thomas ¨Onskog, 08 – 790 84 55.

Means of assistance permitted: Numerical tables and compilation of formulae for this course, Mathematics Handbook (Beta) and Quick guide for TI calculators and pocket calculator.

You should define and explain your notation. Your computations and your line of reasoning should be written down so that they are easy to follow. Numerical values should be given with the precision of two significant digits. The number of exam problems is six. Each problems gives a maximum of ten points. Preliminarly, 24 points will guarantee a passing result. Students with 22-23 points on the exam will be given the grade FX and has the right to complete their exam with an extra examination. Time and location for completion will be announced on the course web page. You must find out yourself if you are eligible for completion.

The exam results will be announced at latest three working weeks after the day of the exam and will be retainable at the student affairs office during a period of seven weeks after the exam.

Uppgift 1

A physician uses the following rule of thumb regarding surgery in case of a certain disease: If she is more than 80% certain that a patient suffers from the disease, she recommends surgery. If she is less certain, she instead recommends further diagnostics, which are expensive and can be painful to the patient. For a specific patient, the doctor is 60% certain that the patient has the disease - based on a number of diagnostics and earlier similar cases - and decides that the patient should undergo test A. The patient is diabetic and test A has the property that it always gives a positive result if the patient has the disease, but for diabetics who do not have the disease it gives a positive result in 30% of the cases. Assume that test A gives a positive result for the patient. What should the doctor do, recommend surgery or carry out further tests? Motivate your answer. (10 p)

Uppgift 2

The trendy bathing store Poolen och Plurret wants to sell out the 2 000 remaining bathing-trunks from this years’ collection at the end of the season. To this end, the store offers a sale on the remaining bathing-trunks. Based on the experience from earlier sales, it is known that the number of bathing-trunks that any customer buys at a sale is independent of the number of trunks bought by other customers and that it can be considered to be a random variable X with the probability function

p_{X}(k) =

0.2 om k = 0, 0.5 om k = 1, 0.2 om k = 2, 0.1 om k = 3, 0 om k > 3.

forts Tentamen i SF1901 2017-08-14 2

Calculate using a suitable and well motivated approximation the smallest number of customers that have to visit the store in order for the probability to be 90% that all the remaining bathing-

trunks are sold. (10 p)

Uppgift 3

In so-called extreme value models (of interest in the insurance industry) one sometimes uses a continuous random variable X with the distribution function

F_{X}(x) =

1 − 1

x^{2}, f¨or x ≥ 1,

0, annars.

a) Calculate, for every real number x the probability

G(x) = P (X ≤ 5 + x|X > 5) .

(7 p) b) Is G(x) defined in part (a) a distribution function? Motivate your answer thoroughly. To acquire points in part (b), it is required that the correct answer is provided on part (a). (3 p)

Uppgift 4

An airline claims to be more punctual than competing airlines operating at a certain large airport.

In this context, “Punctuality” is measured as the fraction of airplanes that depart no more than 20 minutes after the stated time of departure. The airline says that less than one out of 15 of their flights are delayed more than 20 minutes and, if true, this would confirm the airline’s claim of being more punctual than their competitors. The airline has recently signed an agreement with a large furniture company, which gives the airline monopoly on the business flights of the employees of the furniture company and gives the furniture company reduced ticket prices in return. However, the board of the furniture company doubts that the claim of the airline regarding its punctuality is correct and asks its employees to register, during the coming month, if their flights from the airport depart in time, that is within 20 minutes from the stated time of departure, or not. Out of the 250 flights that the employees of the furniture company undertake during this month, 20 are delayed more than the admissible 20 minutes.

Carry out a statistical test on the (possibly approximate) significance level 5% to test the airline’s punctuality claim. Carefully state your hypotheses and motivate your conclusions. (10 p)

Uppgift 5

The trendy bathing store Poolen och Plurret sells bathing-trunks, which are available in four different colours. To be able to plan the production, the store manager wants to know if the relative popularity of the four different colours is approximately the same as last year or not. This is investigated as follows. In the beginning of the bathing season, the colours of the 2 000 bathing- trunks that are first sold are registered. If the proportions of the four colours are similar to the proportions of the 2 000 bathing-trunks first sold in the beginning of last year’s bathing season, the production is not changed. If, on the other hand, the proportions differ to a large enough extent, the production is altered to match the sales in this year’s investigation. The table below shows the number of bathing-trunks sold in each colour in this year’s and last year’s investigations.

forts Tentamen i SF1901 2017-08-14 3 Colour Blue Black Green Red

Last year 943 357 498 202 This year 860 347 476 317

Based on the data above, decide if the production should be altered or not. Use the significance level 5 %. Carefully state your hypotheses and motivate your conclutions. (10 p)

Uppgift 6

In signal processing one often uses a method known as dithering. A very simple and rudimentary example of dithering is as folllows. Let m be an unknown number (’an analogue signal’), which is known to satisfy 0 ≤ m ≤ 1. The random variable X is uniformly distributed on (0, 1). We construct a new random variable (dithering of m)

Y = bm + Xc,

where bxc is the so-called integer part, that is the largest integer being smaller than or equal to x.

a) According to this construction, Y is a binary random variable in the sense that it has two possible values 0 och 1. Determine the probability function for Y . (5 p) b) Let

y_{1} = 1, y_{2} = 1, y_{3} = 1, y_{4} = 0, y_{5} = 1, y_{6} = 0, y_{7} = 1, y_{8} = 0

be random zeros and ones, which can be seen as samples from Y1, Y2, . . . , Y8, respectively,
where Y_{i} = bm + X_{i}c and all X_{i} are independent, U (0, 1)-distributed random variables.

Estimate m based on y_{1}, y_{2}, . . . , y_{8} using the maximum likelihood method (ML).

(Students that have not solved part (a) may use the probability function P (Y = 0) = 1 − p,

P (Y = 1) = p, 0 ≤ p ≤ 1 and estimate p using ML.) (5 p)

Good luck!