This Exam P sample reference tests Exponential Distribution. For an exponential loss, the expected excess above a deductible equals the mean times the survival probability at that deductible. Applying that identity gives a mean of approximately 1790.190, which matches choice B.
How to solve this Exponential Distribution question
Setup
Setup
Represent the insurer's payment as the positive part of the loss after the deductible.
d=0.4β,Y=(X−d)+
Model
Model
Use the exponential survival function and memoryless property to factor the expected payment into the chance of crossing the deductible and the mean residual loss.
E[Y]=Pr(X>d)E[X−d∣X>d]
Pr(X>d)=e−d/β=e−0.4,E[X−d∣X>d]=β
Compute
Compute
Equate the stop-loss expectation to the supplied expected benefit and solve for the exponential mean.
1200=βe−0.4
β=1200e0.4=1790.189637…
Answer
Answer
The nearest listed amount is 1790.
β≈1790.190(B)
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AMultiplying 1200 by the survival factor gives 1200 exp(−0.4)=804.38; the factor must divide the expected payment when solving for the mean.
CUsing 1-0.4=0.6 as though it were an exponential survival probability gives 1200/0.6=2000, but the correct survival value is exp(−0.4).
DDividing the correct 1790.19 result by a further 0.6 gives 2983.65; that applies an unsupported retained-loss fraction after the deductible has already been handled.
EUsing the probability of falling below the deductible gives 1200/(1-exp(−0.4))=3639.89, which conditions on the wrong side of the threshold.
Original practice · fully worked
Original variant: environmental cleanup reimbursement
An environmental response fund reimburses 80% of the portion of a chemical-cleanup cost above a trigger. The cleanup cost C is exponential with mean θ, the trigger is 0.2 θ, and the expected reimbursement per incident is 1000 dollars. Determine θ.
A 819
B 1250
C 1527
D 1908
E 6896
Variant answer in brief
Coinsurance multiplies the exponential excess-loss expectation by 0.8. Solving the resulting equation gives θ approximately 1526.753, so choice C is correct.
Setup
Setup
Write the fund's reimbursement using the trigger and the 80% reimbursement rate.
R=0.8(C−0.2θ)+
Model
Model
For an exponential cost, the expected excess above 0.2 times the mean is the mean multiplied by the corresponding survival factor.
E[R]=0.8θe−0.2
Compute
Compute
Set the modeled reimbursement equal to 1000 and isolate the mean.
1000=0.8θe−0.2
θ=0.81000e0.2=1526.753448…
Answer
Answer
Rounding to the listed whole-dollar values selects choice C.
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