Worked solution · Exam P
This Exam P sample reference tests Joint Distributions. Grouping the nine joint outcomes by whether their maximum is zero, one, or two gives masses 0.38, 0.37, and 0.25. The resulting expected maximum is 0.87, so choice C is correct.
Independent solution
Setup
Let M be the larger of the two payments and collect joint cells that produce the same value.
Model
A maximum of one comes from the three nonzero cells in the upper-left two-by-two block. Every cell containing a two contributes to the remaining mass.
Compute
Weight each possible maximum by its probability.
Answer
The expected larger payment is 0.87 in the table's units.
Test the method on unseen problems
30 original multivariate random variables problems with complete solutions and wrong-choice analysis. Written for Exam P, not copied from the SOA set. One-time $19 PDF.
Error diagnosis
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
Original practice · fully worked
A diagnostic system records a daily peak severity M in {0,1,2,3}. Historical tail probabilities are P(M≥1)=0.72, P(M≥2)=0.35, and P(M≥3)=0.10. Calculate E[M].
A bounded nonnegative integer variable has mean equal to the sum of its positive tail probabilities. Adding the three supplied tails gives 1.17, so choice D is correct.
Setup
Represent the severity as the number of positive thresholds it crosses.
Model
Take expectations term by term.
Compute
Substitute the three tail probabilities.
Answer
The expected peak severity is 1.17.
Keep practicing
These references share syllabus concepts or sit nearby in the official sequence. All 140 Exam P multivariate random variables practice problems.
Drill the method: Exam P Tail Integral Formula and Expected Value problems
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