Worked solution · Exam P
This Exam P sample reference tests Order Statistics. For three independent Uniform(0,10) observations, the maximum has distribution function (x/10)³. Integrating its density gives expected maximum 3(10)/4 = 7.5, which is choice B.
Independent solution
Setup
Let M be the largest of the three independent uniform observations.
Model
The maximum does not exceed x exactly when all three observations do not exceed x.
Compute
Integrate x against the maximum density.
Answer
The expected largest observation is 7.5.
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.
Error diagnosis
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Original practice · fully worked
Four independent calibration readings are uniformly distributed between 60 and 90. A technician records only the largest reading. Calculate its expected value.
After scaling to Uniform(0,1), the maximum of four observations has mean 4/5. Mapping back to [60,90] gives 60 + 30(4/5) = 84, so choice D.
Setup
Standardize each reading to a unit-uniform variable.
Model
The maximum of four unit-uniform observations has density four times u cubed.
Compute
Find the standardized expected maximum and transform it back to the original scale.
Answer
The expected strongest calibration reading is 84.
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 Uniform Distribution problems
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