This Exam P sample reference tests Poisson Distribution. Two adjacent Poisson modes at 2 and 3 force the associated rate to equal 3, and the stated mean ratio therefore makes the other rate 12. A Poisson distribution with integer rate 12 has modes 11 and 12, so the verified answer is choice D.
Use the ratio of consecutive Poisson probabilities to characterize when neighboring counts have equal maximum probability.
pk=e−λk!λk,pk−1pk=kλ
Model
Model
For the distribution whose two modes are 2 and 3, those two masses must be equal.
p2p3=3λlow=1
λlow=3
Compute
Compute
Apply the four-to-one mean relationship, recalling that a Poisson mean equals its rate. For an integer rate n, the equal largest masses occur at n-1 and n.
λhigh=4λlow=4(3)=12
Modes(Poisson(12))={11,12}
Answer
Answer
The two modal counts are 11 and 12, which are listed together under choice D.
11 and 12(D)
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These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThe pair 8 and 9 would be the modes for rate 9, but that rate violates the required four-to-one relationship because the lower rate is 3.
BThe values 8 and 12 come from multiplying the two original modal counts 2 and 3 by four; modal counts do not scale linearly with the mean.
CThe value 10 is the midpoint of the incorrectly scaled values 8 and 12, not a Poisson mode obtained from the rate.
EThe value 12 is the new mean and is one mode, but an integer Poisson rate also makes the preceding count 11 equally likely.
Original practice · fully worked
Original variant: network packet-failure mode
A coastal monitoring network models its hourly packet-failure count N with a Poisson distribution. During calibration, engineers determine that the probability of exactly 7 failures is three halves of the probability of exactly 6 failures. Determine the mode or modes of N.
A 6 only
B 7 only
C 9 only
D 10 only
E 10 and 11
Variant answer in brief
The consecutive-mass ratio is λ divided by 7, so the calibration relationship gives λ equal to 10.5. A noninteger Poisson rate has the unique mode floor(10.5) = 10, making D correct.
Setup
Setup
Write the ratio of the two consecutive Poisson masses.
Pr(N=6)Pr(N=7)=e−λλ6/6!e−λλ7/7!=7λ
Model
Model
Set the theoretical ratio equal to the observed calibration ratio.
7λ=23
λ=10.5
Compute
Compute
Because the rate is not an integer, the Poisson mass increases through floor(λ) and decreases afterward.
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.
