This Exam P sample reference tests Exponential Distribution. The expected-payment coefficient is the first-year failure probability plus half the probability of failure in years two and three. Solving gives 5,644.23, choice D.
How to solve this Exponential Distribution question
Setup
Setup
Use the exponential cumulative distribution at years one and three to identify the probabilities of the two payment layers.
F(t)=1−e−t/10
Model
Model
The contract pays the full amount for first-year failure and half the amount for failure during years two or three. Set their probability-weighted sum equal to 1,000.
1000=x{F(1)+0.5[F(3)−F(1)]}
Compute
Compute
Dividing the target expectation by the combined payment coefficient gives approximately 5,644.227.
x=1−e−0.1+0.5(e−0.1−e−0.3)1000=5644.227
Answer
Answer
The benefit amount is approximately 5,644, selecting choice D.
5644(D)
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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
Variant answer in brief
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.
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