This Exam P sample reference tests Exponential Distribution. The expected refund per printer combines a full-refund first-year probability and half-refund second-year probability; for 100 units it is 10,255.90, choice D.
How to solve this Exponential Distribution question
Setup
Setup
The lifetime distribution is exponential with mean two, so its cumulative distribution gives the probability of failure by each warranty boundary.
F(t)=1−e−t/2
Model
Model
A first-year failure receives the full refund, while a failure during the second year receives half. These disjoint payment layers must be weighted separately.
E[R]=200F(1)+100(F(2)−F(1))
Compute
Compute
Multiplying the expected refund per printer by 100 gives 10,255.90, which rounds to 10,256.
100E[R]=20000(1−e−1/2)+10000(e−1/2−e−1)=10255.90
Answer
Answer
The expected refund for 100 printers is approximately 10,256, selecting choice D.
10,256(D)
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EThe value 12,642 gives every failure in the first two years the full refund. It fails to reduce the refund by half for second-year failures.
Original practice · fully worked
Original variant: expected linearly declining early-failure credit
A component lifetime T is exponential with mean 4 years. If failure occurs before year 2, a service credit of 200(1-T/2) is issued; otherwise the credit is zero. Find the expected service credit.
A 24.08
B 36.79
C 42.61
D 50.00
E 78.69
Variant answer in brief
Integrating the declining credit against the exponential density over its positive range gives 42.612, choice C.
Setup
Setup
The service credit is positive only for failure times below two and declines linearly from 200 to zero over that interval.
B(T)=200(1−T/2)1{0<T<2}
Model
Model
Compute the expectation by integrating the time-dependent credit against the exponential lifetime density over the interval where the credit is positive.
E[B]=∫02200(1−t/2)41e−t/4dt
Compute
Compute
Evaluating the integral gives an expected credit of approximately 42.612.
E[B]=42.612264
Answer
Answer
The expected declining service credit is approximately 42.61, selecting choice C.
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