Worked solution · Exam P
This Exam P sample reference tests Sums of Independent Random Variables. The independent components have combined mean 250 and combined standard deviation √(40²+30²)=50. Dividing 50 by 250 gives coefficient of variation 0.20, so choice A is correct.
Independent solution
Setup
Add the two component means to obtain the expected combined loss.
Model
Independence makes the component variances additive.
Compute
Form the dimensionless ratio of standard deviation to mean.
Answer
The combined coefficient of variation is 0.20.
Test the method on unseen problems
30 original multivariate 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
Two independent laboratory costs A and B have means 40 and 60. The standard deviation of A is 6, while the coefficient of variation of A+B is 0.10. Calculate the standard deviation of B.
The combined mean is 100, so CV 0.10 implies combined standard deviation 10 and variance 100. Independence gives Var(B)=100-36=64, hence SD(B)=8 and choice C is correct.
Setup
Use the combined mean and coefficient of variation to recover the total standard deviation.
Model
Independent component variances add to the total variance.
Compute
Isolate the unknown variance and take its square root.
Answer
The unknown component standard deviation is 8.
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