Question

Q1: An electrical system consists of 2 components (C1 and C2) functioning independently, and are set in a serial layout. The failure time of each component follows an exponential distribution with rate 2 in every year. Thus the system will operate if both components function, and it will fail if any one component fails. (a) Obtain the failure density and distribution of the system (b) Obtain the survival function/distribution of the system. (c) Based on your answer in (a) and (b), calculate i) the probability that the system fails in less than 8 months. ii) the expected time to failure of the system.Q1: An electrical system consists of 2 components (C1 and C2) functioning independently, and are set in a serial layout. The failure time of each component follows an exponential distribution with rate 2 in every year. Thus the system will operate if both components function, and it will fail if any one component fails. (a) Obtain the failure density and distribution of the system (b) Obtain the survival function/distribution of the system. (c) Based on your answer in (a) and (b), calculate i) the probability that the system fails in less than 8 months. ii) the expected time to failure of the system.Q1: An electrical system consists of 2 components (C1 and C2) functioning independently, and are set in a serial layout. The failure time of each component follows an exponential distribution with rate 2 in every year. Thus the system will operate if both components function, and it will fail if any one component fails. (a) Obtain the failure density and distribution of the system (b) Obtain the survival function/distribution of the system. (c) Based on your answer in (a) and (b), calculate i) the probability that the system fails in less than 8 months. ii) the expected time to failure of the system.

Answer 1

Answer:-

Given That:-

An electrical system consists of 2 components (C1 and C2) functioning independently, and are set in a serial layout. The failure time of each component follows an exponential distribution with rate 2 in every year. Thus the system will operate if both components function, and it will fail if any one component fails.

(a) Obtain the failure density and distribution of the system.

Given,

The distribution function of Y is

F(y) = P(Y y)

= P(min() y)

= 1 - P(min() > y)

Since are independent.

= - P()

= 1 -

= 1 -

So,

And the density function of y is

(b) Obtain the survival function/distribution of the system.

The survival function of the system is

= 1 - F(y)

=

So

(c) Based on your answer in (a) and (b), calculate

i) the probability that the system fails in less than 8 months

The probability that system will fail in less than 8 months is given by

Since 8 months = 8/12 months

= 1 - 0.695.

= 0.9305

ii) the expected time to failure of the system.

The expected time to failure of the system is

Now using integration by parts we get

E(Y) = 0.25

So the expected tome to failure of the system is 0.25 year.

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