Worked solution · Exam P
This Exam P sample reference tests Beta-Ratio Moments. For Y=X/(1-X), direct beta integrals give E[Y]=2 and E[Y²]=10. Therefore Var(Y)=10-4=6, choice C.
Independent solution
Setup
Substituting the odds transformation into the density reduces the first two moments to the displayed beta-type integrals.
Model
The first integral gives mean two and the second gives second raw moment ten.
Compute
Variance is the second raw moment minus the squared mean, so the result is six.
Answer
The transformed ratio has variance 6.
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
A random proportion X has density f(x)=60x²(1-x)³ for 0<x<1. Define Y=X/(1-X). Calculate Var(Y).
Integrating the transformed moments gives E[Y]=1 and E[Y²]=2. Hence Var(Y)=2-1²=1, choice B.
Setup
Apply the odds transformation to the stated beta-type density and form its first two raw moments.
Model
The transformed mean is one and the transformed second raw moment is two.
Compute
Subtracting the squared mean from the second raw moment gives variance one.
Answer
The odds transformation has variance 1.
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