Worked solution · Exam P
This Exam P sample reference tests Covariance and Correlation. Writing the common marginal variance as v and the covariance as c gives v+c=5 and 5v-4c=16. Solving yields v=4 and c=1, so choice D is correct.
Independent solution
Setup
Represent the equal marginal variances by v and the unknown covariance by c.
Model
Expand both stated linear-combination variances, retaining the covariance cross terms.
Compute
Substitute v=5-c into the second equation.
Answer
The covariance is 1.0.
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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 centered calibration readings U and V have variances 9 and 16. A diagnostic score is D=2U-V, and repeated trials show Var(D)=36. Calculate Cov(U,V).
Expanding Var(2U-V) gives 4(9)+16-4Cov(U,V)=36. Therefore Cov(U,V)=4, so choice C is correct.
Setup
Let c denote the covariance of the two calibration readings.
Model
Expand the variance of the weighted difference, including its cross term.
Compute
Insert the three supplied variances and isolate c.
Answer
The two readings have covariance 4.
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