This Exam P sample reference tests Mixture Distributions. Weight the zero, total-loss, and partial-loss payment components. Their combined expected payment is about 328, selecting choice B.
Measure payments in thousands. Partial damage pays the amount of loss above the 1-thousand deductible, total loss pays 14, and the remaining state pays zero.
P(partial)=0.04,P(total)=0.02
payment=max(X−1,0)for partial damage, in thousands
Model
Model
Take expectations within the mutually exclusive partial- and total-loss branches, then weight those conditional payments by their branch probabilities.
E[C]=0.02(14)+0.04∫115(x−1)(0.5003)e−x/2dx
Compute
Compute
The total-loss branch contributes 0.02(14)=0.28. The deductible-adjusted partial-loss integral is 1.2049345, contributing 0.04(1.2049345)=0.0481974.
∫115(x−1)(0.5003)e−x/2dx≈1.2049345
E[C]≈0.28+0.04(1.2049345)=0.3281974
Answer
Answer
The combined expected payment is 0.3281974 thousand, or approximately 328, so choice B is selected.
328(B)
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AThe value 320 results from assigning a payment of 1 to every partial-loss event: 0.02(14)+0.04(1)=0.32 thousand. The partial payment is variable, not identically 1.
Original practice · fully worked
Original variant: maintenance reserve with three repair states
A laboratory instrument has no repair in 70% of years, a minor repair costing 2 thousand dollars in 20%, and a major repair costing 9 thousand dollars in 10%. A service contract pays the amount above a 1-thousand-dollar threshold. Find its expected annual payment.
A 0.90 thousand
C 1.20 thousand
B 1.40 thousand
D 1.60 thousand
E 2.20 thousand
Variant answer in brief
The contract pays 1 on a minor repair and 8 on a major repair, so the weighted mean is 0.20(1)+0.10(8)=1.40 thousand.
Setup
Setup
Apply the 1-thousand threshold to each repair state. The resulting contract payments are 0, 1, and 8 thousand in the no-, minor-, and major-repair states.
C=(R−1)+
C∈{0,1,8}with probabilities 0.70,0.20,0.10
Model
Model
Because the three repair states are mutually exclusive, the annual expected payment is their probability-weighted payment average.
E[C]=0.70(0)+0.20(1)+0.10(8)
Compute
Compute
Minor repairs contribute 0.20(1)=0.20 thousand and major repairs contribute 0.10(8)=0.80 thousand; no-repair years contribute zero.
E[C]=0.20+0.80=1.40
Answer
Answer
Thus the expected annual contract payment is 1.40 thousand dollars, corresponding to choice B.
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