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

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 \( 1 / \lambda \). The mean residual life, denoted as \( L(t) \), is defined as \[ L(t)=E[T-t \mid T \geq t]=\int_{t}^{\infty}(x-t) f_{T \mid T \geq t}(x) d x \] where \( f_{T \mid T \geq t}(x) \) is the conditional probability density function given than \( T \geq t \). Note that \( f_{T \mid T \geq t}(x) \) can be expressed as the ratio of the marginal pdf (or regular pdf) \( f(t) \) and reliability function \( R(t) \) as follow \[ f_{T \mid T \geq t}(x)=\frac{f(x)}{P(T \geq t)}=\frac{f(x)}{R(t)} \] Where \[ R(t)=P\{T \geq t\} \] Answer the following question: (1) Drive \( \mathrm{R}(\mathrm{t}) \) ? (2) \( f_{T \mid T \geq t}(x) \) ? (3) Compute L(t)? (4) If \( \lambda=0.1 \) failure/year, and \( t=5 \) years, what is the value of \( L(t=5) \) ?