This Exam P sample reference tests Poisson Distribution. Partition the Poisson count into the four payout categories, then compute both the first and second payout moments. Subtracting the squared mean gives a variance of approximately 138,861, which rounds to choice B.
Let N denote the count and C the resulting payment. Express C as a discrete transformation of N so that its moments can be calculated category by category.
C=⎩⎨⎧0,200,500,1000,N=0,N=1,N=2,N≥3.
Model
Model
For a Poisson rate of 2, the first two positive masses are equal and the upper category is obtained by complement.
p0=e−2,p1=2e−2,p2=2e−2
p3+=1−(p0+p1+p2)=1−5e−2
Compute
Compute
Evaluate the two raw moments using the same probability categories, then apply the variance identity.
E[C]=200p1+500p2+1000p3+=512.7929803…
E[C2]=2002p1+5002p2+10002p3+=401818.0481…
Var(C)=E[C2]−E[C]2=138861.4074…
Answer
Answer
Rounding the variance to the requested thousand selects the second answer choice.
Var(C)≈139000(B)
Continue without hunting through PDFs
All 718 Exam P sample solutions in syllabus order
The searchable 3108-page manual includes this complete solution, its error analysis, and one original worked variant for every active reference.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AMoving the highest payout category from three-or-more events to four-or-more gives about 110,917, a low result in A's range caused by dropping the entire N=3 mass.
CA value near 166,000 results if the correct second moment is paired with an understated mean near 486. The first and second moments must come from the same payout mapping.
DOmitting the 500 payment attached to N=2 while retaining the other categories gives approximately 191,676, which rounds to choice D.
EInflating the highest payment to 1200 raises the variance to about 210,623, pulling the result toward E; the top category pays 1000 in the reference calculation.
Original practice · fully worked
Original variant: cloud incident reserve
A cloud service models its weekly number of severity-one incidents by a Poisson random variable with mean 1.5. Its incident reserve is 300 dollars for exactly one incident, 700 dollars for exactly two, 1200 dollars for three or more, and zero otherwise. Calculate the variance of the weekly reserve.
A 128,015
B 151,755
C 172,846
D 234,156
E 428,384
Variant answer in brief
Compute the category probabilities for a Poisson rate of 1.5 and use them for both reserve moments. The resulting variance is approximately 172,846, so choice C is correct.
Setup
Setup
Let R be the reserve and partition the incident count into zero, one, two, and at least three.
R=⎩⎨⎧0,300,700,1200,N=0,N=1,N=2,N≥3.
Model
Model
Use the Poisson mass function for the first three counts and a complement for the final category.
p0=e−1.5,p1=1.5e−1.5,p2=21.52e−1.5
p3+=1−e−1.5(1+1.5+21.52)
Compute
Compute
Form the two moments and subtract the squared mean.
E[R]=300p1+700p2+1200p3+=505.5073765…
E[R2]=3002p1+7002p2+12002p3+=428383.6364…
Var(R)=428383.6364…−(505.5073765…)2=172845.9287…
Answer
Answer
The variance rounds to the value shown in choice C.
The 3108-page Probability Proof Manual reorganizes 718 verified Exam P solutions by syllabus skill and adds formula proofs, error patterns, and original worked practice.
