This Exam P sample reference tests Deductibles. The payment (X-250)+ has mean about 520.83 and second moment about 434,028. Their difference in squares gives standard deviation about 403.44, choice B.
For a loss uniform from 0 to 1500, the payment is the amount above the 250 deductible, with zero paid below it. Its standard deviation requires the first two raw moments.
X∼Unif(0,1500)
Y=(X−250)+
Model
Model
Integrate the first and second powers of the excess x-250 over losses above the deductible, retaining the zero-payment mass implicitly.
E[Y]=15001∫2501500(x−250)dx
E[Y2]=15001∫2501500(x−250)2dx
Compute
Compute
The payment's first moment is 520.8333 and its second raw moment is 434027.7778. Centering the second moment and taking the square root gives 403.4358.
E[Y]=520.8333
E[Y2]=434027.7778
SD(Y)=E[Y2]−E[Y]2=403.4358
Answer
Answer
The payment standard deviation rounds to 403, which is choice B.
403(B)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThe value 361 is the rounded standard deviation 1250 divided by the square root of 12 for Y conditional on a positive payment. It omits the zero-payment mass P(X ≤ 250)=1/6.
CThe value 433 is the original uniform loss standard deviation, approximately 1500 divided by the square root of 12. It ignores the deductible entirely.
EThe value 521 is the rounded mean payment 520.8333, not its standard deviation.
Original practice · fully worked
Original variant: spread of a discrete service reimbursement
A machine repair costs 400 dollars in half of years, 900 dollars in three tenths of years, and 1600 dollars otherwise. A contract reimburses the portion above 500 dollars. Find the reimbursement standard deviation.
A 324.04 dollars
B 388.84 dollars
E 417.61 dollars
D 486.21 dollars
C 520.00 dollars
Variant answer in brief
Payments are 0, 400, and 1100. Their mean is 340 and second moment is 290000, giving standard deviation 417.61.
Setup
Setup
After the 500-dollar threshold, the three repair-cost states produce reimbursements 0, 400, and 1100 with probabilities 0.5, 0.3, and 0.2.
Y=(R−500)+
Y∈{0,400,1100}
Model
Model
Compute the first two reimbursement moments from those three states before forming the centered second moment.
E[Y]=0.5(0)+0.3(400)+0.2(1100)=340
E[Y2]=0.3(4002)+0.2(11002)=290000
Compute
Compute
With first moment 340 and second raw moment 290000, subtracting the squared mean and taking the square root gives standard deviation 417.6123.
SD(Y)=290000−3402=417.6123
Answer
Answer
Therefore the reimbursement standard deviation is 417.61 dollars, corresponding to choice E.
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