A system consists of five identical components connected in series as shown: 2 3 -5 As soon as one components fails, the entire system will fail. Suppose each component has a lifetime that is exponentially distributed with = 0.01 and that components fail independently of one another. Define events A, = {/th component lasts at least thours),-1,..., 5, so that the A's 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 A₁ Ag? OA₂UA₂ A₂ UA4 As OA₂ A₂UA₂ A₂ UAS од, пад пазпад пад O A₁ UA₂ U A3 UA4 UAS (b) Using the independence of the A,'s, compute P(X 2 t). P(X z 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 1. O X is an exponential distribution with 20.05. O X is a gamma distribution with parameters a = 1 and 0.05. O X is an exponential distribution with 2-1. (c) Suppose there are n components, each having exponential lifetime with parameter A. What type of distribution does X have? O X is a gamma distribution with parameters a 1 and 1/2. O X is an exponential distribution with parameter = e. O X is a gamma distribution with parameters a- A and B - n. O X is an exponential distribution with parameter nå.
A system consists of five identical components connected in series as shown: 2 3 -5 As soon as one components fails, the entire system will fail. Suppose each component has a lifetime that is exponentially distributed with = 0.01 and that components fail independently of one another. Define events A, = {/th component lasts at least thours),-1,..., 5, so that the A's 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 A₁ Ag? OA₂UA₂ A₂ UA4 As OA₂ A₂UA₂ A₂ UAS од, пад пазпад пад O A₁ UA₂ U A3 UA4 UAS (b) Using the independence of the A,'s, compute P(X 2 t). P(X z 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 1. O X is an exponential distribution with 20.05. O X is a gamma distribution with parameters a = 1 and 0.05. O X is an exponential distribution with 2-1. (c) Suppose there are n components, each having exponential lifetime with parameter A. What type of distribution does X have? O X is a gamma distribution with parameters a 1 and 1/2. O X is an exponential distribution with parameter = e. O X is a gamma distribution with parameters a- A and B - n. O X is an exponential distribution with parameter nå.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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P17
Transcribed Image Text:A system consists of five identical components connected in series as shown:
□
| 4 |-
5
As soon as one components fails, the entire system will fail. Suppose each component has a lifetime that is exponentially distributed with = 0.01 and that components fail independently of one another. Define events A, = {ith component lasts at least thours}, i = 1, ..., 5, so
that the A's 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 A₁,..., A5?
O A₁UA₂ A3 UA4 A5
O A₁ MA₂ UA3 A4 UA5
O A₁ A₂ A3 A4 A5
O A ₁ U A₂ U A 3 UA4 UA5
(b) Using the independence of the A,'s, compute P(X 2 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 X is a gamma distribution with parameters α = 0 and 3 = 1.
O X is an exponential distribution with λ = 0.05.
O X is a gamma distribution with parameters a = 1 and 3 = 0.05.
O X is an exponential distribution with λ = 1.
(c) Suppose there are n components, each having exponential lifetime with parameter 1. What type of distribution does X have?
O X is a gamma distribution with parameters a = 1 and 3 = 1/2.
O X is an exponential distribution with parameter λ = e.
O X is a gamma distribution with parameters a = λ and ß = n.
O X is an exponential distribution with parameter nå.
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