Probability · multivariate diagnostic PDF
Exam P Multivariate & Risk Models: 30 Worked Problems
Build control over joint densities, conditioning, covariance, sums, order statistics, and normal approximations with complete five-choice diagnostics.
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- New practice problems
- 30
- Pages
- 76
- Time limit
- 180 min
- Price
- $19
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One original Exam P practice question with full solution
The marginal variances of two calibration errors are Var(X)=4 and Var(Y)=25. Their correlation is not specified and may take any feasible value from -1 through 1. An index is defined as Z=3X-2Y+7. Calculate the largest possible value of Var(Z).
- 16
- 100
- 120
- 136
- 256
Writing Cov(X,Y)=10rho gives Var(Z)=136-120rho. This is largest at rho=-1, where the variance is 256, so choice E.
Setup
Express covariance through the unknown correlation and the two standard deviations.
Model
Expand the variance, retaining the signed covariance cross term.
Compute
The expression decreases with correlation, so choose the smallest feasible correlation.
Answer
The largest possible variance is 256.
Why the other choices are wrong
- A
- The value 16 is the largest possible standard deviation, 3(2)+2(5). The question asks for variance, which is 16 squared.
- B
- The value 100 is (-2) squared times Var(Y). It keeps only the Y contribution and omits X and covariance.
- C
- The value 120 is the greatest possible magnitude of the covariance cross term. It omits both marginal variance terms.
- D
- The value 136 is 9(4)+4(25), obtained by setting correlation to zero. Independence is not given, and zero correlation does not maximize this expression.
Independent answer check
| Primary value | 256 |
|---|---|
| Primary method | Expand the correlated linear-combination variance as a function of rho and optimize it over [-1,1]. |
| Independent value | 256 |
| Independent method | The standard-deviation bound for 3X-2Y is 3SD(X)+2SD(Y)=6+10=16. Perfect negative correlation between X and Y makes the two weighted random parts move together, attaining variance 16 squared=256. |
| Answer | E · agreement passed |
Contents
Inside the 76-page edition
Candidates preparing in English for the Probability (P) Exam
Table of contents
- 30 original multivariate problems
- Joint, marginal, and conditional models
- Covariance and linear combinations
- Order statistics and independent sums
- Central limit theorem and error diagnostics
What is included
30 verified original variants selected without duplication for this focused diagnostic. Official problem wording is not included.
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Every included result is reproducible
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Questions about this edition
Are these official sample questions?
No. Every question is an independently written original variant. Official problem wording is not reproduced.
Are complete solutions included?
Yes. Every problem includes Setup, Model, Compute, and Answer steps plus a concrete explanation for each of the four incorrect choices.
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Probability Multivariate & Risk Models Practice Workbook
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