Division of Mathematical Statistics
EXAM IN SF1900 PROBABILITY THEORY AND STATISTICS, WEDNESDAY OCTOBER 25, 2017 AT 08.00–13.00.
Examiner: Camilla Land´en, 08 – 790 61 97.
Means of assistance permitted: Numerical tables and compilation of formulae for this course, Mathematics Handbook (Beta) 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 at least 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 right to complete their exam. Time and location for completion will be announced on the course web page. You must find out yourself if you are eligible for completion.
Points from the quiz exam and computer workshop during the first Autumn half semester will be accounted in accordance with the examination rules of the course.
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.
There are three possible types of error that can occur on a certain type of product: A, B and C, respectively. Errors of type A occur independently of errors of types B and C. Moreover, the probabilities that errors A, B and C occur are 0.05, 0.02 and 0.03, respectively. A product that has errors of type B also has errors of type C with probability 0.5. A product is considered to be defect if it has at least one of the error types. What is the probability that a randomly chosen
product is defect? (10 p)
The illumination system in a large reception hall consists of a large number of units that break down independently of each other. The time that elapses from the breakdown of one unit to the breakdown of the next unit is exponentially distributed with expectation 0.5 days.
a) Assume that 6 units have broken down. Determine the probability that it takes more than one day before another unit breaks down, that is until there are 7 broken units in the
illumination system. (2 p)
b) The company that is responsible for the illumination in the reception hall replaces broken units only after 100 units have broken down. Determine the probability that it takes more
than 45 days before broken units are replaced. (8 p)
Please turn over!
cont. exam in SF1900 2017-10-25 2
The survival function of a random variable X is defined as SX(x) = 1 − P (X ≤ x). Let x1, . . . , xn be a sample of the independent random variables X1, . . . , Xn, all having the survival function SX(x) = 1 − xθ+1, where 0 ≤ x ≤ 1 and θ > −1. Determine the least-square estimate of the
unknown parameter θ. (10 p)
A scientist at an Agricultural school wants to investigate if the content of quercetine, an antioxidant considered to prevent cancer, differs between two kinds of apples. She measures the quercetine content (in mg/100 g) in 8 apples of the first kind (x1, . . . , x8) and in 10 apples of the second kind (y1, . . . , y10) and calculates x = 18P8
i=1xi = 4.4723, P8
i=1(xi− x)2 = 0.1277, y = 101 P10 i=1yi = 4.6266, and P10
i=1(yi − y)2 = 0.1997. The measurements can be considered to be observations of independent random variables X1, . . . , X8 and Y1, . . . , Y10. It is assumed that Xi ∈ N (µ1, σ), i = 1, ..., 8, and Yi ∈ N (µ2, σ), i = 1, ..., 10, where the parameters µ1, µ2 and σ are unknown.
a) Determine a 95% confidence interval for µ1 − µ2 and test on level of significance 5% the hypothesis H0: µ1 = µ2. The conclusion should be clearly stated and motivated. (5 p) b) Now assume, in contrast to in the (a)-part, that it is known from previous studies that σ = 0.16. Determine a 95% confidence interval for µ1 − µ2 and test on level of significance 5% the hypothesis H0: µ1 = µ2. The conclusion should be clearly stated and motivated. (5 p)
Sm˚att & gott is a chocolate sweet consisting of milk chocolate draped with coloured caramel. The company that manifactures the sweets claims that the occurence of the different colours are as in the table below:
Brown Yellow Red Orange Blue Green
30% 20% 20% 10% 10% 10%
A large bag of Sm˚att & gott-sweets was randomly chosen and turned out to contain 58 brown, 45 yellow, 27 red, 14 orange, 12 green and 13 blue sweets. Perform an appropriate test to determine if this result supports the claim of the company. State your hypotheses and motivate your conclusions
carefully. Use significance level 5%. (10 p)
An astronomer in Sweden wants to measure the distance µ to a bright star. She carries out n1 me- asurements, x1, . . . , xn1, considered to be samples of independent random variables X1, . . . , Xn1. A colleague in Chile wants to measure the same distance and conducts n2 measurements, y1, . . . , yn2, considered to be samples of independent random variables Y1, . . . , Yn2. The measurements of the Swedish instrument have standard deviation σ1 and the measurements of the Chilean instrument have standard deviation σ2. None of the instruments are biased. The astronomers want to combine their estimates of the distance, ¯x and ¯y, using the linear combination µ∗obs = c1· ¯x + c2· ¯y, where c1 and c2 are two constants. State the condition on the constants c1 and c2 that makes the estimate µ∗ unbiased. Assume further that n1 = n2 and that σ1 = 2 · σ2. What values of the constants c1 and c2 gives the most efficient estimate µ∗obs = c1· ¯x + c2· ¯y? (10 p)