Worked solution · Exam P
This Exam P sample reference tests Continuous Distributions. Centering at 6 turns the supplied density into (3/2)u² on (-1,1). Symmetry gives zero centered mean, and integrating u² against that density gives 3/5=0.60, so choice B is correct.
Independent solution
Setup
Shift the variable to the midpoint of its support by setting U=X-6.
Model
The centered density is even, so U has mean zero. Translation does not change variance.
Compute
Integrate the centered second moment over the two-sided unit interval.
Answer
The variance is 0.60 square units.
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
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
Original practice · fully worked
A calibrated instrument has symmetric error U, measured in units. For R=|U|, the calibration report gives P(R≤r)=r⁴⁄¹⁶ for 0≤r≤2. Calculate the mean squared error E[U²].
Differentiating the absolute-error distribution gives f_R(r)=r³⁄⁴. Because U²=R², its expectation is the integral of r⁵⁄⁴ from 0 to 2, which equals 8/3 and selects choice D.
Setup
Work directly with the nonnegative absolute error R, since squaring removes the sign of U.
Model
Differentiate the reported distribution function and form the second raw moment of R.
Compute
Evaluate the one-dimensional moment integral.
Answer
The mean squared calibration error is 8/3.
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 Variance problems
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