Original practice · fully worked
Original variant: calibrate a continuously declining early-failure benefit
A machine lifetime T is exponential with mean 8 years. For failure before year 3, a contract pays x(1-T/3); it pays zero later. Determine x if the expected payment must equal 500.
- A
2500 - B
2800 - C
3010 - D
3486 - E
4000
Answer and calculation
Integrating the declining benefit factor against the exponential density gives coefficient 0.166105. Dividing 500 by it gives 3,010.15, choice C.
Setup
Setup
The declining payment is positive only for failure before year three and decreases linearly from the full amount to zero.
B(T)=x(1−T/3)1{0<T<3} Model
Model
Integrate the payment factor against the exponential density over the three-year payment interval, then multiply by the unknown scale.
500=x∫03(1−t/3)81e−t/8dt Compute
Compute
The integral coefficient is approximately 0.166105. Dividing 500 by this coefficient gives approximately 3,010.149.
x=500/0.1661047434=3010.149 Answer
Answer
The required payment scale is approximately 3,010, selecting choice C.
3010(C) 