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.
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.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 67E
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Transcribed Image Text: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 exponentially
distributed with 1 = 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,,
O A, U A2 U A3U AqU A5
O A, U A, NA, U A NA5
O A, N A, U A3 NAUAS
O 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 distribution
does X have?
O xis a gamma distribution with parameters a = 1 and B = 1/2.
%3D
O 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.
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