Problem 2: Exponential distribution is often used to estimate the mean Residual Life of a product...

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Problem 2: Exponential distribution is often used to estimate the mean Residual Life of a product or system. For instance, a car has been used for 5 years, and how many additional years can be used with no major failure. The expected additional lifetime given that a system has survived until time t is called the mean residual life. Let T represent the exponential life of a system, and its mean lifetime is 1/2. The mean residual life, denoted as L(t), is defined as L(t) = E[T-t | T≥t] =(x-t) frrt (x) dx where f(x) is the conditional probability density function given than Izt. Note that f(x) can be expressed as the ratio of the marginal pdf (or regular pdf) f(t) and reliability function R(t) as follow Where frTz(x) = f(x) _f(x) P(T≥t) R(t) R(t) = P{T≥t} Answer the following question: (1) Drive R(t)? (2) fT\Tzt (x)? (2.1) (2.2) (2.3) (3) Compute L(t)? (4) If λ=0.1 failure/year, and t-5 years, what is the value of L(t=5)? Problem 2: Exponential distribution is often used to estimate the mean Residual Life of a product or system. For instance, a car has been used for 5 years, and how many additional years can be used with no major failure. The expected additional lifetime given that a system has survived until time t is called the mean residual life. Let T represent the exponential life of a system, and its mean lifetime is 1/2. The mean residual life, denoted as L(t), is defined as L(t) = E[T-t | T≥t] =(x-t) frrt (x) dx where f(x) is the conditional probability density function given than Izt. Note that f(x) can be expressed as the ratio of the marginal pdf (or regular pdf) f(t) and reliability function R(t) as follow Where frTz(x) = f(x) _f(x) P(T≥t) R(t) R(t) = P{T≥t} Answer the following question: (1) Drive R(t)? (2) fT\Tzt (x)? (2.1) (2.2) (2.3) (3) Compute L(t)? (4) If λ=0.1 failure/year, and t-5 years, what is the value of L(t=5)?

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Let s solve these questions as illustrated Given that 1 Lt ... View the full answer