An alternative model used in reliability treats the environmental stress as a random variable \(X\), with probability density \(f(x)\), and the strength of the component to withstand this stress as an independent random variable \(Y\) having probability density \(g(y)\). Then, the reliability is defined as

\[\begin{aligned} R=P[Y>X] & =\int_{-\infty}^{\infty} \int_{-\infty}^{y} f(x) g(y) d x d y \\ & =\int_{-\infty}^{\infty} F(y) g(y) d y \end{aligned}\]

where \(F(x)\) is the distribution function of \(X\). Evaluate this reliability when

(a) \(X\) has an exponential distribution with \(\alpha=0.01\) and \(Y\) has an exponential distribution with failure rate 0.005 ;

(b) \(X\) has an exponential distribution with \(\alpha=0.005\) and \(Y\) has an exponential distribution with failure rate 0.005 ;

(c) \(\ln X\) has a normal distribution with \(\mu=60\) and \(\sigma=5\) and \(\ln Y\) has a normal distribution with \(\mu=\) 80 and \(\sigma=5\).