Worked solution · Exam P
This Exam P sample reference tests Continuous Random Variables. Normalizing the density gives the loss CDF 1-(1-x)⁴. Its median is about 0.159104 million, below the payment limit, so choice A is correct.
Independent solution
Setup
Normalize the polynomial density on its unit interval.
Model
Integrating gives the loss CDF. The probability accumulated below the payment ceiling already exceeds one half, so the ceiling does not change the median.
Compute
Solve the median equation and convert millions to ordinary currency units.
Answer
The nearest listed median payment is 159,000.
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 server response time T, in seconds, is exponential with mean 4. A monitoring panel records only the excess above one second, so its displayed value is W=max(T-1,0). Calculate the median of W.
Only 22.1% of readings are floored at zero, so the median remains in the positive part. Subtracting one from the exponential median gives 4ln(2)-1=1.773, so choice C is correct.
Setup
Check whether the floor at zero contains at least half of the probability.
Model
Above the floor, W is the increasing transformation T-1. Its median is therefore the raw-time median shifted down by one second.
Compute
Set the transformed CDF equal to one half.
Answer
The display has median 1.773 seconds.
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