This Exam P sample reference tests Exponential Distribution. This is an expected excess-loss calculation for an exponential severity above an ordinary deductible. The payment expectation equals the survival probability at two times the mean residual loss, giving exp(−2), so choice A is correct.
How to solve this Exponential Distribution question
Setup
Setup
Let L be the loss amount and express the insurer's payment under an ordinary deductible of two.
L∼Exp(1),P=(L−2)+
Model
Model
A payment occurs only when the loss crosses the deductible. Conditional on that event, exponential memorylessness makes the excess have the original mean of one.
Pr(L>2)=e−2
E[L−2∣L>2]=1
Compute
Compute
Multiply the chance of a positive payment by its conditional mean amount.
E[P]=Pr(L>2)E[L−2∣L>2]
E[P]=e−2(1)=e−2=0.1353352832…
Answer
Answer
The expected payment per loss is e to the power negative two.
e−2(A)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
BMultiplying the deductible amount two by the probability of exceeding it gives 2 exp(−2); the payment is the excess above the deductible, not the deductible itself.
CThis uses survival beyond one rather than beyond two, effectively replacing the stated deductible by the distribution's mean.
DThe quantity 3 exp(−2) is E[L times the indicator that L exceeds 2], which includes the retained first two units and therefore does not subtract the deductible.
EThis is the mean loss before applying any deductible and consequently overstates the insurer's expected payment.
Original practice · fully worked
Original variant: capped equipment-repair layer
Equipment damage severity L is exponentially distributed with mean 2 repair units. A service contract pays the portion of a loss lying between 1 and 4 units: it pays nothing below 1, L-1 between 1 and 4, and a maximum of 3 units above 4. Calculate the expected contract payment per incident.
A 0.1353
B 0.4712
C 0.6065
D 0.9424
E 1.2131
Variant answer in brief
The capped layer can be valued by integrating its payment tail from zero to three. This gives 2(exp(−1/2)-exp(−2))=0.942391, so choice D is correct.
Setup
Setup
Represent the contract as a three-unit layer attaching when the severity reaches one.
L∼Exp(1/2)
W=min((L−1)+,3)
Model
Model
For a nonnegative capped payment, integrate the probability that the payment exceeds each level from zero up to its cap.
E[W]=∫03Pr(W>t)dt
Pr(W>t)=Pr(L>1+t)=e−(1+t)/2
Compute
Compute
Evaluate the payment-tail integral across the full three-unit layer.
E[W]=∫03e−(1+t)/2dt
E[W]=2(e−1/2−e−2)=0.9423907530…
Answer
Answer
The expected capped-layer payment is approximately 0.9424 repair units.
The 3108-page Probability Proof Manual reorganizes 718 verified Exam P solutions by syllabus skill and adds formula proofs, error patterns, and original worked practice.
