Reliability and Survival Analysis Questions and Answers

Answer: D

\(F_{\alpha}\) is IFR if \(1 \le \alpha\).

This is the Weibull distribution. Its hazard rate is

\[ \begin{aligned} h(t) &= \frac{f(t)}{1 - F(t)} \\ &= \alpha\lambda^{\alpha}t^{\alpha - 1} \end{aligned} \]

which is increasing in \(t\) when \(\alpha \gt 1\), constant when \(\alpha = 1\) (the exponential case, which counts as both IFR and DFR), and decreasing when \(\alpha \lt 1\). So the distribution is IFR for \(\alpha \ge 1\) and DFR for \(\alpha \le 1\).

It is not IFR or DFR for every \(\alpha\), so A and B fail, and C has the range reversed.

Answer: C

The reliability is \(1 - (F(t))^{K}\).

A parallel system works as long as at least one unit works, so it fails only when all K units have failed by time \(t\). With independent units each failing by \(t\) with probability \(F(t)\), that probability is \(F(t)^{K}\), and the reliability is its complement.

\((1 - F(t))^{K}\) is the reliability of a series system, which needs every unit to survive. \(F(t)^{K}\) is the failure probability of the parallel system, and \(1 - (1 - F(t))^{K}\) is the failure probability of the series system.