Worked solution · Exam P
This Exam P sample reference tests Continuous Random Variables. Integrating the density gives cumulative distribution 1 minus exp(-4x squared). Setting this equal to one-half yields a median of 0.416277, which rounds to choice C.
Independent solution
Setup
Integrate the density from zero to a positive argument to obtain its cumulative distribution.
Model
A continuous median m places one-half of the probability at or below m.
Compute
Isolate the exponential term, take logarithms, and retain the positive root because the support is positive.
Answer
The median rounds to 0.416.
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
The distance R, in kilometers, from an ocean sensor to a detected acoustic event has density 18r exp(-9r squared) for positive r and zero otherwise. Determine the 75th percentile of the detection distance.
The cumulative distribution is 1 minus exp(-9r squared). Equating it to 0.75 gives a percentile of 0.392470 kilometer, selecting choice C.
Setup
Integrate the density over its positive support.
Model
Let q be the 75th percentile and set its cumulative probability to 0.75.
Compute
Solve the exponential equation for the positive distance.
Answer
The 75th-percentile detection radius is approximately 0.392 kilometer.
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