Verified solution · Exam P
This Exam P sample reference tests Marginal Distributions. Summing the joint mass over the second coordinate gives marginal probabilities 0.1, 0.3, and 0.6 at the positive values of X. The resulting first and second moments are 2.5 and 6.7, so the variance is 0.45 and choice A.
Independent solution
Setup
Marginalize the supplied joint mass function over every allowed value of the second coordinate.
Model
The finite arithmetic sum reduces to a simple marginal mass function.
Compute
Calculate both raw moments from the marginal probabilities.
Answer
The marginal variance is 0.45.
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Original practice · fully worked
A quality team records the number A of priority tags on a production lot and whether a manual review B is requested. Their joint probabilities are P(A=0,B=0)=0.10, P(A=0,B=1)=0.05, P(A=1,B=0)=0.20, P(A=1,B=1)=0.25, P(A=2,B=0)=0.15, and P(A=2,B=1)=0.25. Calculate the variance of A.
Marginalizing over the review indicator gives probabilities 0.15, 0.45, and 0.40 for A = 0, 1, and 2. The first two raw moments are 1.25 and 2.05, so the variance is 0.4875 and choice C.
Setup
Add the two joint cells in each row to obtain the distribution of the tag count.
Model
Use the first and second raw moments of this three-point marginal distribution.
Compute
Evaluate the moments and subtract the squared mean.
Answer
The priority-tag count has variance 0.4875.
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ActuaryProof. “SOA Exam P Sample Question #652 — Verified Solution.” Verified 2026-07-30.
Source reference: official SOA sample-question PDF. Last reviewed 2026-07-30.
