A system consists of five identical components connected in series as shown:

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_i = {ith component lasts at least t hours}, i = 1, . . . , 5, so that the A_is 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 ? t} is equivalent to what event involving A₁, . . . , A₅?

A₁ ? A₂ ? A₃ ? A₄ ? A₅

A₁ ? A₂ ? A₃ ? A₄ ? A₅

A₁ ? A₂ ? A₃ ? A₄ ? A₅

A₁ ? A₂ ? A₃ ? A₄ ? A₅

(b) Using the independence of the A_i's, compute P(X ? t). P(X ? t) =

Obtain F(t) = P(X ? t). F(t) =

Obtain the pdf of X. f(t) =

What type of distribution does X have?X is a gamma distribution with parameters ? = 0 and ? = 1.X is an exponential distribution with ? = 0.05. X is a gamma distribution with parameters ? = 1 and ? = 0.05.X is an exponential distribution with ? = 1. (c) Suppose there are n components, each having exponential lifetime with parameter ?. What type of distribution does X have?X is a gamma distribution with parameters ? = 1 and ? = 1/?.X is an exponential distribution with parameter ? = e. X is a gamma distribution with parameters ? = ? and ? = n.X is an exponential distribution with parameter n?.

A system consists of five identical components connected in series as shown: 3 5 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 A1, . . . , As? O AnA2 nA3 nAnA5 O A, UA, UA3 UAQUA 5 O A, UA, nA 3 UAAnA 5 O AnA, UA 3 nAQUAS (b) Using the independence of the A;'s, compute P(X > t). P ( X > 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 B = 1. O X is an exponential distribution with 1 = 0.05. O X is a gamma distribution with parameters a = 1 and B = 0.05. O X is an exponential distribution with 1 = 1.(c) Suppose there are n components, each having exponential lifetime with parameter 51. What type of distribution does X have? 0 Xis a gamma distribution with parameters a = l and ,8 = 1M. 0 Xis an exponential distribution with parameter 1 = e. O Xis a gamma distribution with parameters a: = l and ,8 = n. O Xis an exponential distribution with parameter n1