Original practice · fully worked
Original variant: percentile of a repair reimbursement
A protection plan has no covered event with probability 0.70. Conditional on an event, the loss is exponential with mean 2,000 dollars, and the plan applies a 200-dollar deductible. Find the 91st percentile of the plan payment.
- A
1,833 - B
2,000 - C
2,100 - D
2,208 - E
2,408
Answer and calculation
The 91st unconditional percentile corresponds to the 70th conditional percentile. The loss quantile is -2000 ln(0.30)=2407.95, so payment is 2207.95, choice D.
Setup
Setup
The payment distribution has probability 0.70 at zero. The requested 91st percentile therefore lies within the positive-payment component.
0.70+0.30FL(y+200)=0.91 Model
Model
Rescaling within the covered-event component shows that the associated loss level is the 70th percentile of the exponential severity.
FL(y+200)=0.70,e−(y+200)/2000=0.30 Compute
Compute
The loss percentile is approximately 2,408. Subtracting the 200 deductible gives a payment percentile of approximately 2,208.
y=−2000ln(0.30)−200=2207.946 Answer
Answer
The payment percentile rounds to 2,208 dollars.
2208(D) 