A system consists of five identical components connected in series as shown:T-2H 3-As soon as one components fails, the entire system will fail. Suppose each component has a lifetime that is exponentiallydistributed with 1 = 0.01 and that components fail independently of one another. Define events A, = {ith component lasts atleast t hours}, i = 1, . .., 5, so that the As are independent events. Let x = the time at which the system fails-that is, theshortest (minimum) lifetime among the five components.(a) The event {X 2 t} is equivalent to what event involving A,,O A, U A2 U A3U AqU A5O A, U A, NA, U A NA5O A, N A, U A3 NAUASO A, N A, N Az n ANA5(b) Using the independence of the A,'s, compute P(X > t).P(X 2 t) =Obtain F(t) = P(X < t).F(t) =Obtain the pdf of X.f(t) =What type of distribution does X have?O xis a gamma distribution with parameters a = 0 and B = 1.O xis an exponential distribution with 2 = 0.05.O xis a gamma distribution with parameters a = 1 and B = 0.05.O xis an exponential distribution with 2 = 1.(c) Suppose there are n components, each having exponential lifetime with parameter 1. What type of distributiondoes X have?O xis a gamma distribution with parameters a = 1 and B = 1/2.%3DO x is an exponential distribution with parameter 1 = e.O x is a gamma distribution with parameters a = 1 and ß = n.O x is an exponential distribution with parameter nå.

Five identical components are connected in series. The components fail independently of each other.…

A system consists of five identical components connected in series as shown: T-2H 3 4 As soon as one components fails, the entire system will fail. Suppose each component has a lifetime that is exponentially distributed with 2 = 0.01 and that components fail independently of one another. Define events A, = {ith component lasts at least t hours}, i = 1, ..., 5, so that the As are independent events. Let X = the time at which the system fails-that is, the shortest (minimum) lifetime among the five components. (a) The event {X 2 t} is equivalent to what event involving A,, ..., A,? O A, U A, U Az U Ag U A5 O A, U Az n Ag U Ag n As O A, N Az U Ag nAgU As O A, N Az n Ag n Ag n As (b) Using the independence of the A,'s, compute P(X 2 t). P(X 2 t) = Obtain F(t) = P(X s t). F(t) = Obtain the pdf of X. f(t) = What type of distribution does X have? O x is a gamma distribution with parameters a = 0 and ß = 1. O x is an exponential distribution with 2 = 0.05. O x is a gamma distribution with parameters a = 1 and ß = 0.05. O x is an exponential distribution with 2 = 1. (c) Suppose there are n components, each having exponential lifetime with parameter 2. What type of distribution does X have? O x is a gamma distribution with parameters a = 1 and B = 1/2. O x is an exponential distribution with parameter 2 = e. O x is a gamma distribution with parameters a = 1 and B = n. X is an exponential distribution with parameter nå.

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...