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4.43 A reliability system has five operating components in parallel. The lifetimes of the components are independent random variables, each having an exponential distribution with expected value. The system fails as soon as only two components are working. What are the expected value and the standard deviation of the time until system failure? Hint: Verify first that min (X, Y) is exponentially distributed with parameter a + 8 when the random variables X and Y are independent and exponentially distributed with parameters a and B.

4.43 A reliability system has five operating components in parallel. The lifetimes of the components are independent random variables, each having an exponential distribution with expected value. The system fails as soon as only two components are working. What are the expected value and the standard deviation of the time until system failure? Hint: Verify first that min (X, Y) is exponentially distributed with parameter a + 8 when the random variables X and Y are independent and exponentially distributed with parameters a and B.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.3: Special Probability Density Functions
Problem 46E
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Questions number 4.43

15 30 89 30 air Problems 45 Figure 4.4. Events of a Poisson process with rate 1 in the time interval (0,45). 1991. Unpredictable events such as shark attacks can be modeled by a Poisson process, with its bursty behavior. This means that there are periods with a much higher than average number of attacks, as well as periods with no attacks at all. 111564-5 Iliw ti woled hamon sti dong SOLAT MAL e to viilidedown an 4.43 A reliability system has five operating components in parallel. The lifetimes of the components are independent random variables, each having an exponential distribution with expected value. The system fails as soon as only two components are working. What are the expected value and the standard deviation of the time until system failure? Hint: Verify first that min (X, Y) is exponentially distributed with parameter a + ß when the random variables X and Y are independent and exponentially distributed with parameters a and B. 4.44 Limousines depart from the railway station to the airport from the early morning till late at night. The limousines leave from the railway station with independent interdeparture times that are exponentially distributed with an expected value of 20 minutes. Suppose you plan to arrive at the railway station at 3 o'clock in the afternoon. What are the exped value and the standard deviation of your waiting time at the an te betwe Sven betwe ni be and 4.47 You in spee ned exp I can TOVO sec ca m 4.48 T e sd T 4.49
Transcribed Image Text:15 30 89 30 air Problems 45 Figure 4.4. Events of a Poisson process with rate 1 in the time interval (0,45). 1991. Unpredictable events such as shark attacks can be modeled by a Poisson process, with its bursty behavior. This means that there are periods with a much higher than average number of attacks, as well as periods with no attacks at all. 111564-5 Iliw ti woled hamon sti dong SOLAT MAL e to viilidedown an 4.43 A reliability system has five operating components in parallel. The lifetimes of the components are independent random variables, each having an exponential distribution with expected value. The system fails as soon as only two components are working. What are the expected value and the standard deviation of the time until system failure? Hint: Verify first that min (X, Y) is exponentially distributed with parameter a + ß when the random variables X and Y are independent and exponentially distributed with parameters a and B. 4.44 Limousines depart from the railway station to the airport from the early morning till late at night. The limousines leave from the railway station with independent interdeparture times that are exponentially distributed with an expected value of 20 minutes. Suppose you plan to arrive at the railway station at 3 o'clock in the afternoon. What are the exped value and the standard deviation of your waiting time at the an te betwe Sven betwe ni be and 4.47 You in spee ned exp I can TOVO sec ca m 4.48 T e sd T 4.49
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