Worked solution · Exam P
This Exam P sample reference tests Conditional Expectation. The joint expectation of the advanced count times the early-stage condition indicator is 6(0.1)(1-0.8⁵)=0.403392. Dividing by P(at least one early)=1-0.8⁶ gives 0.546708, choice C.
Independent solution
Setup
Let X count early-stage selections, Y count advanced-stage selections, and A be the event X≥1.
Model
For each patient to contribute to Y while A occurs, that patient must be advanced and at least one of the other five must be early-stage.
Compute
Evaluate the joint expectation and normalize by the conditioning probability.
Answer
The conditional expected advanced-stage count rounds to 0.547.
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.
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Original practice · fully worked
Independent event counts A and B are Poisson with means 2 and 3, respectively. Given that their combined count equals 4, calculate the conditional expected value of A.
Conditioned on a combined count of four, the type-A count is Binomial(4,2/5). Its expectation is 4(2/5)=1.6, choice B.
Setup
Combine the two independent Poisson streams.
Model
Given the total, each event is classified as type A with probability proportional to its rate.
Compute
Apply the binomial mean formula.
Answer
The conditional expected type-A count is 1.6.
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