This Exam P sample reference tests Marginal Distributions. Only 40% of the second-period loss remains unreimbursed, while the first-period loss is fully covered. Marginalizing the joint table gives a second-period mean of 0.90, so the expected unreimbursed amount is 0.36 and choice A.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
BThe value 0.76 is 40% of the first-period mean 1.90. That period is fully covered, so it should not create any unreimbursed loss.
CThe value 0.90 is the second-period expected loss before applying its 40% uncovered share.
DThis amount exceeds the entire second-period mean of 0.90, so it cannot be the expectation of only 40% of that nonnegative loss; it reflects a misread table margin or coverage rate.
EThe value 2.80 is the expected total loss from both periods before either reimbursement provision is applied.
Original practice · fully worked
Original variant: festival vendor credit
A theatre festival vendor may incur a booth charge B and an electrical surcharge E. A sponsor pays all of B and 70% of E. Joint probabilities for columns B=0, 4, 10 are: row E=0 has 0.40, 0.15, 0.05; row E=5 has 0.10, 0.08, 0.02; and row E=12 has 0.08, 0.07, 0.05. Calculate the vendor's expected amount not paid by the sponsor.
A 0.30
B 0.40
C 1.02
D 1.74
E 3.40
Variant answer in brief
The surcharge marginal probabilities are 0.60, 0.20, and 0.20 at 0, 5, and 12. Its mean is 3.40, of which the vendor pays 30%, giving 1.02 and choice C.
Setup
Setup
Sum across booth-charge columns to obtain the marginal distribution of the electrical surcharge.
Pr(E=0)=0.40+0.15+0.05=0.60
Pr(E=5)=0.10+0.08+0.02=0.20
Pr(E=12)=0.08+0.07+0.05=0.20
Model
Model
The booth charge is fully sponsored. The vendor's unpaid amount is therefore 30% of the electrical surcharge.
V=0.30E
E[V]=0.30E[E]
Compute
Compute
Evaluate the surcharge mean and then its uncovered portion.
E[E]=0(0.60)+5(0.20)+12(0.20)=3.40
E[V]=0.30(3.40)=1.02
Answer
Answer
The vendor expects to pay 1.02 credits beyond the sponsor's contribution.
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