This Exam P sample reference tests Payment Transformations. A reimbursement below 115 corresponds to cost below 150, conditional on cost exceeding 20. The exponential conditional CDF is 1-exp(-1.3)=0.727, choice B.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
CNo direct unconditional cumulative probability at either cost boundary was identified that produces 0.741.
Original practice · fully worked
Original variant: distribution of a two-tier shipping rebate
Shipping cost X is exponential with mean 50 dollars. A rebate is zero through 10 dollars, pays all cost from 10 through 40, and then pays half of additional cost. Given a positive rebate, find the probability the rebate is below 45 dollars.
D 0.6988
B 0.7534
C 0.7981
A 0.8347
E 0.8647
Variant answer in brief
A 45-dollar rebate corresponds to cost 70. Conditional on X>10, the probability X<70 is 1-exp(-(70-10)/50)=0.6988.
Setup
Setup
A positive rebate corresponds to shipping cost above 10 dollars. Inverting the two-tier schedule shows that a rebate below 45 corresponds to cost below 70.
Y>0⟺X>10
Y<45⟺X<70
Model
Model
Condition on cost above 10 and use exponential memorylessness over the 60-dollar interval from 10 to 70.
P(Y<45∣Y>0)=P(X<70∣X>10)
Compute
Compute
The conditional cumulative probability is 1 minus exp(-60/50), equal to 0.698806.
P=1−e−(70−10)/50=1−e−1.2=0.698806
Answer
Answer
Thus the conditional probability is 0.6988, corresponding to choice D.
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