This Exam P sample reference tests Exponential Distribution. An exponential variable whose standard deviation is sigma has survival S(x)=exp(-x/sigma). Lowering the threshold by 0.5 sigma multiplies its survival probability by exp(0.5), giving 0.20exp(0.5)=0.3297 and choice A.
How to solve this Exponential Distribution question
Setup
Setup
Let d be the fixed threshold component. For an exponential variable, the mean equals the standard deviation, so the rate is 1/sigma.
S(x)=Pr(L>x)=e−x/σ
S(d+σ)=0.20
Model
Model
Relate the desired survival probability to the known one by taking their ratio.
S(d+σ)S(d+0.5σ)=e−(d+σ)/σe−(d+0.5σ)/σ
S(d+σ)S(d+0.5σ)=e0.5
Compute
Compute
Multiply the supplied tail probability by the exponential factor for a half-standard-deviation reduction in the threshold.
S(d+0.5σ)=0.20e0.5
S(d+0.5σ)=0.3297442541…
Answer
Answer
The requested probability rounds to 0.33.
0.33(A)
Test the method on unseen problems
Probability Univariate Distributions Practice Workbook
30 original univariate random variables problems with complete solutions and wrong-choice analysis. Written for Exam P, not copied from the SOA set. One-time $19 PDF.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
BThis doubles 0.20 to 0.40 because the additional threshold is halved. Exponential tail probabilities scale by e to a power, not inversely with distance.
CRecovering S(d)=0.20e and then taking its complement gives 1-0.20e=0.4563, which rounds to 0.46. That is a different event at a different threshold.
DThis is approximately 0.20e=0.5437, which lowers the known threshold by a full sigma to d instead of by only 0.5 sigma.
EThis linearly interpolates halfway from 0.20 toward 1 to obtain 0.60. Survival over an exponential interval does not change linearly.
Original practice · fully worked
Original variant: conditional cartridge survival
A cartridge has an exponential time to depletion. By hour 12, 75% of cartridges have depleted. One cartridge is known to be active at hour 9. Calculate the probability that it is still active at hour 12.
A 0.2500
B 0.3536
C 0.5000
D 0.7071
E 0.7500
Answer and calculation
The 12-hour survival probability is 0.25. Memorylessness reduces the conditional event to three more hours, so the probability is S(3)=S(12)⁽³⁄¹²⁾=0.25⁽¹⁄⁴⁾=0.7071 and choice D.
Setup
Setup
Convert the depletion percentage into survival and express it through the exponential rate.
S(12)=1−0.75=0.25
e−12λ=0.25
Model
Model
Given activity at hour 9, exponential memorylessness makes the next three hours the only relevant interval.
Pr(T>12∣T>9)=S(3)=e−3λ
Compute
Compute
Scale the known 12-hour survival exponent down to three hours.
e−3λ=(e−12λ)3/12
S(3)=(0.25)1/4=21=0.7071067812…
Answer
Answer
The conditional survival probability rounds to 0.7071.
