Exam1_Solutions - F' . [le DCT-lE-EE’IES 14: 2E1...

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Unformatted text preview: F' . [le DCT-lE-EE’IES 14: 2E1 University of Missouri Department of Statistics Introduction to Mathematical Statistics : 4710/7710 Examination Se tember 2005 ‘ Instructions: Time: 60 min. I. Attempt all questions. 2. Show all workings: Points will not be awarded unless all workings are show. Question I [l 7} Two items are selected from an assembly line and classed as to whether or not they are superior quality (5*), aVerage quality (a), or inferior quality (0. (:1) Construct a tree die to reresent this two stae ex erirnent. (b) (c) List the sam - le - ints that constitute the events: A: the first item selected is of inferior quality 45-5 J eelet. «cl .4:- 1 B: the quality of each of the items is the same 5 5 J M J :2 .4" 1 C: the quality of the first item exceeds that of the second 5 a. , S-e' , div £- 1 (d) Are events A and s mutual] exclusive? m Are eventsA and C mutuall exclusive? M (e) Give a brief verbal descri ion of the events: eesdéfie ‘IlLU-IE-iey‘ et': DCT-lE-EE’IES 14: 21 F'. [32 Question 2 {15] Let X be a Geometric random variable with parameter p. Then the density for X is f(x) = P(1—P)H :- x= L 2r 3: (a) Show fi-nm first principles that the moment generating function for this distribution is given by: (b) E(x) -“-' 1/ P DCT-lE-EE’IE’IS 14: 21 F'. [33 Question 3 [7] Assume that the probability that the air brakes on a large truck will fail on any particular application of the brakes 0.001. (a) Let X denote the number of brake applications before the first failure of the air brakes, i.e. the waiting time for failure of the air breaks. Write down the robabilit distribution for X . 7C“! f<r> 10”)“ J ,3, em; (on???) ,. 33" . (E n- P"- G Q. 2 E. :t. :5 :2 ED ch 2 F. .':F' (D E! (D H E." Z 1: "I {TI CI _., (D F} ._. ... g ‘7'; 0 DD ‘13 (b) W11 E; (c) Two assumptions are necessary for the distribution in part (a): Independence of the trials and constant probability of success from trial to trial. Are these assum tions realistic in this settin‘? Comment. Neither assumption may be completely satisfied: Because of wear and tear the probability of failure will almost certainly increase with each application of the brakes. slowly at first but more later on as all the parts become worn, till eventually the probability of failure is so great that driving the truck is a danger and the brakes need to be renewed. But this also means that each application of the brakes is affected by the previous applications, and therefore the applications are not even independent. Of course, once failure occurs. no more successes are possible. This is very different to the example of tossing a coin and waiting for the first “failure”. So on close examination, neither assumption is satisfied. However. when new, the geometric distribution may nonetheless still provide a good simple approximation for the waiting time to failure distribution. UCT-lE-EE’IE’IS 14: 21 F'. |Z|4 Question 4 [21] A random telephone poll is conducted to ascertain public opinion concerning the construction of a nuclear power plant in a particular community of 150 000 people, of whom 90 000 are opposed to its construction. Let X denote the number of negative responses obtained in 15 calls. (a) Find the densi _ for X . Hint: Make a sketch! (b) (C) (d) DCT-lE-EE’IE’IS 14: 21 P as Question 5 [8] Let X be a random variable with mean p and variance or2 , Le. X ~ (Mal) and let 2 = £154 he the standardized random variable. Prove that E(Z) = D and D“ V3112) =1 using the theorems flux E and Va: . For 12(2) = 0: For Var(Z)=1: Vai— r (E ) DCT-lE-EEES 14:22 P.EE Question 6 [7] The average number of lightening strikes on transformers during the severe thunderstorm season in Missouri is 2 per week. Assume that a Poisson process is in operationr and find the probability that during the next storm session one must wait at most 1 week in order to see the first transformer strike. END TDTRL P.BE ...
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Exam1_Solutions - F' . [le DCT-lE-EE’IES 14: 2E1...

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