Worked solution · Exam P
This Exam P sample reference tests Sums of Independent Random Variables. This problem asks for the standard deviation of an independent sum. Squaring the four component standard deviations and adding gives variance 30, so the total standard deviation is √(30)=5.4772 and choice B.
Independent solution
Setup
Let S be the aggregate and denote the four independent components by X1 through X4.
Model
Independence makes all pairwise covariance terms zero, so add component variances rather than component standard deviations.
Compute
Evaluate the variance sum and take its square root.
Answer
The total standard deviation rounds to 5.5.
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
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
Original practice · fully worked
Three independent centered calibration errors U, V, and W have standard deviations 1.5, 2, and 4 units. A corrected reading has error R=2U-V+0.5W. Calculate the standard deviation of R.
Variance scales with the square of each coefficient. Independence gives Var(R)=4(1.5²)+2²+0.5²(4²)=17, so SD(R)=√(17)=4.123 and choice B.
Setup
Write the variance rule for the stated linear combination of independent errors.
Model
Convert the three supplied standard deviations to variances before applying the squared coefficients.
Compute
Add the three weighted variance contributions, then return to standard-deviation units.
Answer
The corrected reading has standard deviation approximately 4.123 units.
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