Discussion
Suppose the lifetimes of two systems, system 1 and system 2, follow exponential distributions with failure rates 2 and ½ respectively. Let the probability that system 'i' fails prior to its mean time to failure be denoted by pi, i = 1, 2. Which of the following is true ?
Answer: B
p1 = p2.
For an exponential lifetime with failure rate \(\lambda\), the mean time to failure is \(1/\lambda\), and
\[ \begin{aligned} P\left(T \lt \frac{1}{\lambda}\right) &= 1 - e^{-\lambda \cdot 1/\lambda} \\ &= 1 - e^{-1} \\ &\approx 0.632 \end{aligned} \]
whatever the rate. So both systems have the same probability, about 0.632, of failing before their own mean life; the rates 2 and ½ change the mean but not this probability. Their product is about 0.4, not 1.
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