This Exam P sample reference tests Exponential Distribution. A positive payment occurs with probability exp(-d/10), and memorylessness makes that positive excess exponential with moments 10 and 200. Thus Var(Y)=100(2 exp(-d/10)-exp(-d/5)), choice D.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThe expression 100-d subtracts the deductible directly from the original variance. Variance does not shift linearly under a positive-part payment transformation.
BThe expression (10-d) squared treats the payment as though it were a deterministic difference between the original mean and deductible. It ignores both randomness and the zero-payment mass.
CThe expression 100 exp(-d/10) is E[Var(Y|payment status)]. It includes within-positive-payment variance but omits variation between zero and positive payment states.
EThis replaces the exponential scale-square 100 by (10-d) squared in the correct probability bracket. Memorylessness leaves the positive excess mean at 10, not 10-d.
Original practice · fully worked
Original variant: variance of a coinsured discrete repair payment
A repair loss is 2, 7, or 15 thousand credits with probabilities 0.30, 0.50, and 0.20. A plan pays 60% of the amount above 4 thousand credits, subject to a maximum payment of 6 thousand credits. Calculate the variance of the payment, in squared thousand-credit units.
A 1.80
B 2.10
C 4.41
D 6.00
E 8.82
Variant answer in brief
The three losses produce payments 0, 1.8, and 6 with the stated probabilities. Their mean is 2.1 and second moment is 8.82, so the variance is 8.82-2.1 squared=4.41, choice C.
Setup
Setup
Apply the deductible, coinsurance, and maximum to each possible repair loss.
Y=min(0.60(L−4)+,6)
Y(2)=0,Y(7)=1.8,Y(15)=6
Model
Model
Compute the first two raw moments from the three transformed outcomes.
E[Y]=0.30(0)+0.50(1.8)+0.20(6)=2.10
E[Y2]=0.30(0)+0.50(1.82)+0.20(62)=8.82
Compute
Compute
Subtract the squared mean from the second raw moment.
Var(Y)=8.82−2.102=4.41
Answer
Answer
The payment variance is 4.41 squared thousand-credit units.
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