3044hw2f12sols

# Are integers and x is a geometrically distributed

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are integers and X is a geometrically distributed random variable. The probability of a failure is denoted by q and P.X > s/ D 1 X j D s C 1 q j 1 p D q s ; P.X > t/ D q t ; and P.X > s C t/ D q s C t I so ; P OE.X > s C t/ j X > s ç D q s C t =q s D q t

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3 which is equal to P.X > t/ . 19. Let X be defined as the lifetime of the component. Then X is exponential . D 1= 10,000 hours) with cumulative distribution function F.x/ D 1 e x=10000 ; x > 0 Given that the component has not failed for s D 10; 000 or s D 15; 000 hours, the probability that it lasts 5000 more hours is P.X 5000 C s j X > s/ D P.X 5000/ D :6065 In both cases, this is due to the memoryless property of the exponential distribution. 21. The service times, X i , are exponential . D 1=50/ with cumulative distribution function F.x/ D 1 e x=50 ; x > 0 (a) The probability that two customers are each served within one minute is P.X 1 60; X 2 60/ D OEF.60/ ç 2 D .:6988/ 2 D :4883 (b) The total service time, X 1 C X 2 , of two customers has an Erlang distribution (assuming independence) with cumulative distribution function F.x/ D 1 1 X i D 0 OEe x=50 .x=50/ i =i äç ; x > 0 The probability that the two customers are served within two minutes is P.X 1 C X 2 120/ D F.120/ D :6916 39. Let X be defined as the length of the i th shaft, and Y as the linkage formed by i shafts. Then X i is normally distributed. (a) The linkage, Y , formed by the three shafts is distributed as Y N 3 X i D 1 i ; 3 X i D 1 2 i !
• Spring '08
• ALEXOPOULOS
• Probability theory, Exponential distribution, Qj, Weibull, 000 hours, 10,000 hours

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