Worked solution · Exam P
This Exam P sample reference tests Poisson Distribution. This asks for the most probable count of a Poisson variable with mean 2.5. Adjacent probability ratios show that the mass increases through count 2 and decreases at count 3, so the unique mode is 2 and choice C.
Independent solution
Setup
Write the ratio between consecutive Poisson masses.
Model
The mass function rises while this ratio exceeds one and falls after it drops below one.
Compute
The direction changes after count two, identifying the peak.
Answer
The modal count is two.
Test the method on unseen problems
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.
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Original practice · fully worked
Three independent calibration panels each display an integer selected uniformly from 1 through 5. The controller records M, the median of the three displayed values. Calculate the mode of M.
The median CDF at m is the probability that at least two of three panels are at most m. Differencing it gives masses 13, 31, 37, 31, and 13 out of 125, so the unique mode is 3 and choice C.
Setup
Let q=m/5 be the probability that one panel displays at most m.
Model
The median is at most m exactly when at least two of the three panels are at most m.
Compute
Difference consecutive CDF values and compare the five masses.
Answer
The controller most often records three.
Keep practicing
These references share syllabus concepts or sit nearby in the official sequence. All 387 Exam P univariate random variables practice problems.
Drill the method: Exam P Poisson Distribution problems
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