QUESTION 6 (16 Marks) A continuous random variable X has a Weibull distribution if it has a probability distribution function given by

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1 QUESTION 6 (6 Marks) A continuous random variable X has a Weibull distribution if it has a probability distribution function given by x f ( x) e x / if x otherwise where and are positive constants. The Weibull distribution provides a good model for the length of life of many electronic components and biological plants and animals. It has been found that the length of life (measured in thousands of hours) of an aircraft guidance system follows a Weibull distribution with and. a. Find correct to the nearest hour: (i) The expected value of the length of life of an aircraft guidance system. marks (ii) The standard deviation of the length of life of an aircraft guidance system. marks The School For Excellence 7 Mathematical Methods Examination Questions Page

2 b. Find correct to four decimal places the probability that a randomly selected aircraft guidance system will have a length of life that exceeds four thousand hours. marks c. If three guidance systems are used independently, find correct to four decimal places the probability that exactly one of the three fails prior to four thousand hours of use. marks The School For Excellence 7 Mathematical Methods Examination Questions Page

3 d. A particular aircraft guidance system is known to have already lasted for four thousand hours. Find correct to four decimal places the probability that the guidance system will last for less than five thousand hours. 3 marks The School For Excellence 7 Mathematical Methods Examination Questions Page 3

4 e. Find correct to the nearest hour, the expected length of life of an aircraft guidance system that has already lasted for four thousand hours. marks The School For Excellence 7 Mathematical Methods Examination Questions Page

5 QUESTION 6 Include the yellow highlighted sections in your exam response. a. (i) Probability density function: x E( X ) x e dx f ( x) x M x e x dx 896 Note: The use of an upper integral limit of is a valid approximation for obtaining an answer correct to the required accuracy. The expected value of the length of life of an aircraft guidance system is equal to 8 hours, correct to the nearest hour. A (ii) Var( X ) x x ( x 896) ( x 896) dx dx 68 M sd ( X ) Note: The use of an upper integral limit of is a valid approximation for obtaining an answer correct to the required accuracy. The standard deviation of the length of life of an aircraft guidance system is equal to 6 hours, correct to the nearest hour. A x x b. Pr( X ) dx dx 9 correct to four decimal places. Note: The use of an upper integral limit of is a valid approximation for obtaining an answer correct to the required accuracy. mark for equation to be solved M mark for correct answer A The School For Excellence 7 Mathematical Methods Examination Questions Page

6 c. Let Y denote the random variable number of guidance systems that fail. Then Y ~ Bi ( p , n 3 ) Note: In order to avoid accumulation of rounding error, a greater degree of accuracy than required in the final answer is used during the calculation. Pr( Y ) 3 C ( 7983) ( 7983) 976, correct to four decimal places. Answer is.976 A d. pdf x Pr( X ) Pr( X X ) M Pr( X ) From Part (b): Pr( X ) 896 Pr( X ) x dx Therefore: Pr( X X ) 93, correct to four decimal places. 896 M Answer is.93 A e. A conditional expected value is required: E( X X ) xf ( x X ) dx M Conditional probability density function: f ( x X x ) Pr( X ) x x, X otherwise M The School For Excellence 7 Mathematical Methods Examination Questions Page 6

7 Therefore: E( X X ) x x e 6978 dx mark for equation to be solved M Note: In order to avoid accumulation of rounding error, a greater degree of accuracy than required in the final answer is used during the calculation. The expected length of life of an aircraft guidance system that has already lasted for four thousand hours is equal to hours, correct to the nearest hour. A The School For Excellence 7 Mathematical Methods Examination Questions Page 7