This Exam P sample reference tests Loss Models. A payment below 0.5 is equivalent to loss below C+0.5. The loss CDF is x², so (C+0.5)²=0.64 and C=0.3, choice B.
The payment is zero below the deductible and otherwise equals the excess loss. A payment below 0.5 is therefore equivalent to a loss below the deductible plus 0.5.
Y=(X−C)+,FX(x)=x2(0<x<1)
Model
Model
The loss cumulative distribution is the square of the loss amount on its support, so substitute the shifted threshold into that distribution.
Pr(Y<0.5)=Pr(X<C+0.5)=(C+0.5)2
Compute
Compute
Taking the positive square root gives a loss threshold of 0.8. Subtracting 0.5 leaves deductible 0.3.
(C+0.5)2=0.64,C=0.30
Answer
Answer
The deductible is 0.3, selecting choice B.
0.3(B)
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EThe value 0.8 is the loss threshold corresponding to cumulative probability 0.64. The deductible is 0.5 less than that threshold.
Original practice · fully worked
Original variant: deductible recovered from a uniform payment percentile
A loss amount spreads probability evenly over all values between zero and twelve. A policy pays (X-d) when X>d and zero otherwise. If the probability that the payment is at most 3 equals 0.75, determine d.
A 2
B 3
C 4
D 9
E 6
Variant answer in brief
Payment at most three is equivalent to X at most d+3. Uniformity gives (d+3)/12=0.75, so d=6, choice E.
Setup
Setup
A payment at most three occurs exactly when the uniform loss is at most the deductible plus three.
Y=(X−d)+
Model
Model
For a uniform loss on zero to twelve, that probability is the shifted threshold divided by twelve.
Pr(Y≤3)=Pr(X≤d+3)=12d+3
Compute
Compute
The required loss threshold is nine, so subtracting the payment limit of three gives deductible six.
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