This Exam P sample reference tests Uniform Distribution. A uniform excess payment has mean (1000-d)²⁄²⁰⁰⁰. Setting this to one quarter of 500 yields d=500, choice C.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThe value 250 sets the deductible equal to 25% of the support maximum. The payment ratio is quadratic in the remaining support width, so this does not make the expected payment 25% of E[X].
EThe value 750 follows from treating the expected-payment ratio as linear: the remaining support fraction is set to 0.25. The correct payment ratio is the square of that fraction.
Original practice · fully worked
Original variant: choose a threshold for discrete renewable surplus
Daily renewable output is 200 megawatt-hours on one fifth of days, 500 on one half, and 900 on the remaining three tenths. A storage operator keeps only output above a threshold h between 200 and 500. Choose h so expected stored surplus is 224 megawatt-hours.
A 300 MWh
B 340 MWh
C 350 MWh
D 370 MWh
E 400 MWh
Variant answer in brief
For a threshold between 200 and 500, only the 500 and 900 outcomes contribute. Solving 0.5(500-h)+0.3(900-h)=224 gives h=370.
Setup
Setup
Because 200<h<500, the 200-MWh outcome contributes no stored surplus, while the 500- and 900-MWh outcomes contribute 500-h and 900-h.
X∈{200,500,900}
200<h<500
Model
Model
Weight the two positive surplus amounts by probabilities 0.5 and 0.3 and set their sum equal to the required mean of 224 MWh.
E[(X−h)+]=0.5(500−h)+0.3(900−h)
Compute
Compute
The expectation simplifies to 520-0.8h. Solving 520-0.8h=224 gives h=370, which lies in the assumed interval.
520−0.8h=224
h=370
Answer
Answer
Thus the required storage threshold is 370 MWh, corresponding to choice D.
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