Two components with the same mean life

Topic: Random variables and distributions · more questions on this topic
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Two components each have expected lifetime 3 years. The first lifetime is exponential. The second is normal with standard deviation 1/2 year.

1. Find the probability that the first still works after 4.5 years, given that it works after 4 years.

2. Find the probability that the second still works after 4.5 years, given that it works after 4 years.

Worked answer (click to show)

1. The first lifetime is exponential with rate 1/3. By the memoryless property, surviving 4 years makes no difference:

\mathbb{P}(T>4.5\mid T>4)=\mathbb{P}(T>0.5)=e^{-0.5/3}=0.846.

2. The second lifetime is \operatorname{N}(3,0.25). Standardising, 4 and 4.5 years are 2 and 3 standard deviations above the mean, so

\mathbb{P}(T>4.5\mid T>4)=\frac{1-\Phi(3)}{1-\Phi(2)}=\frac{0.00135}{0.02275}=0.059.

The normal component is worn out: having already lasted far longer than usual, it is very unlikely to last another half year. The exponential component does not age at all.