Worked solution · Exam P
This Exam P sample reference tests Continuous Distributions. The density kernel has derivative exp(-x squared)(1-2x squared). Its interior maximum is at 1/√(2)=0.70711, which rounds to choice C.
Independent solution
Setup
A positive proportionality constant does not affect where the density is largest.
Model
Differentiate the kernel using the product and chain rules.
Compute
Solve the interior stationary equation and verify the derivative changes from positive to negative.
Answer
The mode is approximately 0.71.
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 normalized reliability score X has cumulative distribution F(x)=3x²-2x³ for 0≤x≤1, with F(x)=0 below zero and F(x)=1 above one. Determine the score at which the probability density is largest.
Differentiating the CDF gives density 6x(1-x). Its derivative vanishes at x=1/2 and changes from positive to negative there, so choice C is the mode.
Setup
Differentiate the supplied cumulative distribution on the interior of its support.
Model
Differentiate the density to find its interior stationary point.
Compute
Solve for the critical score and compare it with the zero-density endpoints.
Answer
The reliability density is maximized at 0.500.
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