Avd. Matematisk statistik
EXAM IN SF1901 PROBABILITY THEORY AND STATISTICS, TUESDAY MAY 30, 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(6). Each problems gives a maximum of ten (10) points.
Preliminary, 24 points will guarantee a passing result. The grade FX (the exam can completed by extra examination) is given to students with 22 − 23 points. Time and location for completion will be announced on the course web page. You must find out yourself if you are eligible to completion.
Points from workshops and quiz exams during the second Spring half semester will be accounted in accordance to rules. 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 date of the exam.
Uppgift 1
Waloddi Weibull, who was a Professor at KTH Royal Institute of Technology, made important contributions to the study of strength of materials and rupture in solids. The so-called Weibull distribution is named after him; a random variable X is said to be W(λ, β)-distributed, where λ and β > 0, if
FX(x) =
(1 − e−(λx)β, for x ≥ 0,
0, elsewhere.
A system consists of two components whose life spans are given by independent W(λ, β)-distributed random variables. The system is operating as long as both components are operating. Determine the distribution function for the life span of the system. (10 p)
Uppgift 2
In an ergonomic investigation of a keyboard for work stations, 31 very experienced operators where recruited and given the possibility to determine the most convenient height xi, i = 1, . . . , 31 of the arm rest (as measured in cm from the ground level). The mean observed height 311 (x1+ . . . + x31) was equal to 80.0 cm. We model the 31 observations as independent samples from a normal distribution N (m, 2.0).
Please turn over!
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a) Determine a two-sided confidence interval for m with confidence level 95%. (7 p) b) Test the null hypothesis
H0 : m = 81.5 (cm) against the alternative hypothesis
H1 : m 6= 81.5 (cm).
at significance level 5%. The conclusion regarding H0 must be clearly stated and motiva-
ted. (10 p)
Uppgift 3
The amount of precipitation during one day (24 hours), measured in mm, can be considered to be a random variable X. It holds that
X = Y Z,
where Y och Z are independent random variables with the following properties
P (Y = 0) = 1 − p, P (Y = 1) = p, f¨or 0 ≤ p ≤ 1P (Z > x) = e−x/m, for x > 0, m > 0.
This means that the probability for precipitation is p and if there is precipitation a given day, the amount is given by an Exp(m)-distributed random variable.
a) Determine E(X) and V (X) as functions of p and m. (5 p)
b) Assume that the amounts of precipitation during different days are independent, identically distributed random variables with the same distribution as X above. From precipitation data, we have concluded that E(X) = 1.5 and V (X) = 6.75 (corresponding to m = 3 and p = 1/2 in part a) of the problem). Calculate the probability that the total amount of precipitation during one year (= 365 days) exceeds 500 mm using an appropriate and well
motivated approximation. (5 p)
Uppgift 4
In certain problems in Particle Physics, we encounter the probability density
f (x) =
1
2(1 + θx) −1 ≤ x ≤ 1,
0 elsewhere,
where −1 ≤ θ ≤ 1 is an unknown parameter. Let x1, . . . , xn be samples of independent random variables X1, . . . , Xn, which all have this distribution.
a) Derive the least square estimate of the parameter θ. (5 p) b) Determine if the least square estimate of θ is biased. (5 p)
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Uppgift 5
A group of teachers in Mathematics at a large engineering school developed a new diagnostic test in Mathematics for newly admitted students. The teachers believe that the difficulty level of the diagnostic test is such that 40% of the newly admitted students will fail the test, 40% will recieve one of the grades E, D or C and the rest will receive on of the grades B or A.
A random sample of 200 newly admitted students took the diagnostic test and it turned out that 60 of these students recieved one of the grades E, D or C, that 45 recieved one of the grades B or A and that the rest of the students failed the test.
Conduct a suitable test on the approximative significance level 5% to see if the intended grade
distribution is in line with the result above. (10 p)
Uppgift 6
A die manifacturing machine has been misprogrammed so that it on each of the six sides of a dice randomly paints one of the numbers 1, . . . , 6, i.e. it paints one of the numbers with the same probability regardless of what it has painted on the other sides of the dice. Assume that we take a random dice that has been produced by the misprogrammed machine.
a) What is the probability that we get a six in one toss with the dice? (3 p) b) What is the probability that we get two sixes in two tosses with the dice? (7 p)
Good luck!
