This Exam P sample reference tests Mixed-Distribution Percentiles. Because 80% of policies pay zero, the unconditional 95th percentile is the conditional 75th loss percentile less the deductible: 3000 ln(4)-500=3659, choice B.
How to solve this Mixed-Distribution Percentiles question
Setup
Setup
The payment distribution has probability 0.80 at zero. Because the requested 95th percentile lies above that atom, it must be located within the positive-loss component.
0.80+0.20FL(y+500)=0.95
Model
Model
Removing the zero-payment mass and rescaling the remaining probability shows that the required loss level is the 75th percentile of the exponential severity.
FL(y+500)=0.75,1−e−(y+500)/3000=0.75
Compute
Compute
The 75th-percentile loss is approximately 4,159. Subtracting the 500 deductible gives approximately 3,659.
y=3000ln4−500=3658.883
Answer
Answer
Rounding to the listed whole-dollar amount gives 3659.
3659(B)
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CThe value 4,159 is the conditional loss percentile before the 500 deductible is subtracted.
DThe value 8,487 treats the unconditional 95th percentile as the 95th percentile within the covered-loss component, then subtracts the deductible.
EThe value 8,987 both uses the wrong conditional percentile and fails to subtract the deductible.
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
Variant answer in brief
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.
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