Worked solution · Exam P
This Exam P sample reference tests Covariance of a Sum. The correlation implies covariance 0.64(40)(30)=768. Adding both variances and twice the covariance gives 4036, choice D.
Independent solution
Setup
Convert the given correlation to covariance by multiplying it by the two standard deviations.
Model
The variance of a sum includes both marginal variances and twice the covariance because the cross-product appears in both orders.
Compute
Adding 1,600, 900, and twice 768 gives 4,036.
Answer
The variance of the combined annual count is 4036.
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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 weekly workload measures have variances 900 and 400 and correlation -0.25. Find the variance of their sum.
The covariance is -0.25(30)(20)=-150. Therefore the sum variance is 900+400+2(-150)=1000, choice C.
Setup
The standard deviations are 30 and 20. Multiplying their product by the correlation gives covariance negative 150.
Model
The variance of the sum is the two marginal variances plus twice this negative covariance.
Compute
Starting from 1,300 and subtracting 300 gives a combined variance of 1,000.
Answer
The combined workload variance is 1,000.
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