Original practice · fully worked
Original variant: conditional benefit under exponential mortality
A component already operating for 5 years has an exponential remaining-life model with mean 20 years. A warranty pays 2,000 dollars if it fails before year 12. Find the expected payment conditional on operation through year 5.
- A
393.47 dollars - B
590.62 dollars - C
704.69 dollars - D
786.94 dollars - E
1,409.38 dollars
Answer and calculation
Memorylessness leaves a seven-year horizon. The conditional failure probability is 1-exp(-7/20), so expected payment is about 590.6 dollars.
Setup
Setup
Conditional on operation through year 5, failure before year 12 means failure during the next seven years.
P(T<12∣T>5)=1−e−(12−5)/20 Model
Model
Memorylessness leaves an exponential seven-year failure probability, which is multiplied by the 2000-dollar warranty amount.
E[B∣T>5]=2000(1−e−7/20) Compute
Compute
The seven-year failure probability is 1 minus exp(-7/20), yielding expected payment 590.624 dollars.
E[B∣T>5]=590.624 Answer
Answer
The conditional expected warranty payment is 590.62 dollars, corresponding to choice B.
590.62 dollars(B) 