Exam P topic · 23–30% syllabus weight

Exam P Multivariate Random Variables practice problems

Independently written solutions mapped to this official syllabus topic. Pages appear only after all required checks agree.

Topic orientation

What this topic tests

Multivariate Random Variables studies joint, marginal, and conditional distributions together with covariance, correlation, and sums of random variables. The central question is how dependence changes a calculation that would otherwise look one-dimensional.

Problems commonly ask for a marginal or conditional quantity, a moment of a linear combination, or a distribution obtained from several components. A reliable setup identifies the joint support and preserves covariance terms until independence is justified.

Typical failure modes

  • Dropping covariance without establishing independence
  • Normalizing over the wrong joint or conditional support
  • Adding standard deviations instead of variances

Published ledger

Exam P Multivariate Random Variables practice problems and solutions

Each entry shows its independent calculation record and links to the official source without reproducing protected wording.

  1. SOA Exam P Sample Question #31Multivariate Random Variables · Multinomial DistributionVerified
  2. SOA Exam P Sample Question #40Multivariate Random Variables · Normal Distribution · Mixture DistributionsVerified
  3. SOA Exam P Sample Question #69Multivariate Random Variables · Covariance · Central Limit Theorem · Aggregate SumsVerified
  4. SOA Exam P Sample Question #73Multivariate Random Variables · Independence · Exponential Distribution · Joint ProbabilityVerified
  5. SOA Exam P Sample Question #75Multivariate Random Variables · Covariance · Variance · Linear TransformationsVerified
  6. SOA Exam P Sample Question #76Multivariate Random Variables · Joint Distributions · Marginal Distributions · VarianceVerified
  7. SOA Exam P Sample Question #77Multivariate Random Variables · Variance · Linear Transformations · IndependenceVerified
  8. SOA Exam P Sample Question #79Multivariate Random Variables · Order Statistics · Exponential Distribution · IndependenceVerified
  9. SOA Exam P Sample Question #80Multivariate Random Variables · Covariance · Linear Transformations · MomentsVerified
  10. SOA Exam P Sample Question #82Multivariate Random Variables · Conditional Distributions · Bernoulli Distribution · VarianceVerified
  11. SOA Exam P Sample Question #83Multivariate Random Variables · Conditional Distributions · Joint Distributions · VarianceVerified
  12. SOA Exam P Sample Question #89Multivariate Random Variables · Conditional Distributions · Geometric Distribution · Expected ValueVerified
  13. SOA Exam P Sample Question #95Multivariate Random Variables · Conditional Expectation · Binomial Distribution · Continuous DistributionsVerified
  14. SOA Exam P Sample Question #110Multivariate Random Variables · Joint Probability DecompositionVerified
  15. SOA Exam P Sample Question #114Multivariate Random Variables · Covariance of a SumVerified
  16. SOA Exam P Sample Question #169Multivariate Random Variables · Joint Discrete Distributions · Normalization · Expected ValueVerified
  17. SOA Exam P Sample Question #171Multivariate Random Variables · Variance · Optimization · TransformationsVerified
  18. SOA Exam P Sample Question #172Multivariate Random Variables · Conditional Expectation · Bayes’ Theorem · Binomial DistributionVerified
  19. SOA Exam P Sample Question #173Multivariate Random Variables · Conditional Variance · Joint Probability Functions · Conditional DistributionsVerified
  20. SOA Exam P Sample Question #175Multivariate Random Variables · Normal Distribution · Linear Combinations · Variance of Linear CombinationsVerified
  21. SOA Exam P Sample Question #176Multivariate Random Variables · Convolution · Discrete Random Variables · ModesVerified
  22. SOA Exam P Sample Question #177Multivariate Random Variables · Binomial Distribution · Variance of Linear Combinations · IndependenceVerified
  23. SOA Exam P Sample Question #178Multivariate Random Variables · Conditional Expectation · Multinomial Distribution · Indicator Random VariablesVerified
  24. SOA Exam P Sample Question #227Multivariate Random Variables · Mixture Distributions · Poisson Distribution · Law of Total VarianceVerified
  25. SOA Exam P Sample Question #238Multivariate Random Variables · Geometric Distribution · Joint Distributions · IndependenceVerified
  26. SOA Exam P Sample Question #239Multivariate Random Variables · Joint Cumulative Distribution Functions · Discrete Random Variables · IndependenceVerified
  27. SOA Exam P Sample Question #240Multivariate Random Variables · Central Limit Theorem · Sums of Independent Random VariablesVerified
  28. SOA Exam P Sample Question #241Multivariate Random Variables · Central Limit Theorem · Sums of Independent Random VariablesVerified
  29. SOA Exam P Sample Question #242Multivariate Random Variables · Independence · Variance · Mixture DistributionsVerified
  30. SOA Exam P Sample Question #243Multivariate Random Variables · Joint Distributions · Expectation of a Function · Discrete Random VariablesVerified
  31. SOA Exam P Sample Question #244Multivariate Random Variables · Joint Distributions · Marginal Distributions · Expected ValueVerified
  32. SOA Exam P Sample Question #245Multivariate Random Variables · Conditional Variance · Law of Total Variance · Uniform DistributionVerified
  33. SOA Exam P Sample Question #246Multivariate Random Variables · Poisson Distribution · Conditional Distributions · Binomial DistributionVerified
  34. SOA Exam P Sample Question #247Multivariate Random Variables · Joint Distributions · Marginal Distributions · Variance MinimizationVerified
  35. SOA Exam P Sample Question #248Multivariate Random Variables · Covariance and Correlation · Independence · Functions of Random VariablesVerified
  36. SOA Exam P Sample Question #250Multivariate Random Variables · Normal Distribution · Linear Combinations of Independent Random Variables · StandardizationVerified
  37. SOA Exam P Sample Question #251Multivariate Random Variables · Normal Distribution · Linear Combinations of Independent Random Variables · Tail ProbabilitiesVerified
  38. SOA Exam P Sample Question #252Multivariate Random Variables · Sums of Independent Random Variables · Coefficient of Variation · VarianceVerified
  39. SOA Exam P Sample Question #253Multivariate Random Variables · Variance of Linear Combinations · Independence · Affine TransformationsVerified
  40. SOA Exam P Sample Question #254Multivariate Random Variables · Conditional Distributions · Law of Total Variance · Poisson DistributionVerified
  41. SOA Exam P Sample Question #269Multivariate Random Variables · Normal Distribution · Sums of Independent Random Variables · Variance of Independent SumsVerified
  42. SOA Exam P Sample Question #270Multivariate Random Variables · Variance and Standard Deviation · Sums of Independent Random Variables · Variance of Independent SumsVerified
  43. SOA Exam P Sample Question #284Multivariate Random Variables · Exponential Distribution · Covariance and Correlation · Variance of Linear CombinationsVerified
  44. SOA Exam P Sample Question #290Multivariate Random Variables · Central Limit Theorem · Normal Approximation · Sums of Independent Random VariablesVerified
  45. SOA Exam P Sample Question #291Multivariate Random Variables · Joint Probability Functions · Covariance · Correlation CoefficientVerified
  46. SOA Exam P Sample Question #292Multivariate Random Variables · Joint Distributions · Marginal Distributions · VarianceVerified
  47. SOA Exam P Sample Question #293Multivariate Random Variables · Joint Distributions · Expectation of a Function · VarianceVerified
  48. SOA Exam P Sample Question #298Multivariate Random Variables · Central Limit Theorem · Sums of Independent Random Variables · Insurance PricingVerified
  49. SOA Exam P Sample Question #301Multivariate Random Variables · Covariance · Variance of SumsVerified
  50. SOA Exam P Sample Question #302Multivariate Random Variables · Poisson Distribution · Sums of Independent Random Variables · Cumulative ProbabilitiesVerified
  51. SOA Exam P Sample Question #305Multivariate Random Variables · Poisson Distribution · Normal Distribution · IndependenceVerified
  52. SOA Exam P Sample Question #311Multivariate Random Variables · Joint Probability Functions · Variance of Linear Combinations · CovarianceVerified
  53. SOA Exam P Sample Question #315Multivariate Random Variables · Normal Distribution · Transformations of Random Variables · IndependenceVerified
  54. SOA Exam P Sample Question #320Multivariate Random Variables · Central Limit Theorem · Sums of Independent Random VariablesVerified
  55. SOA Exam P Sample Question #324Multivariate Random Variables · Poisson Distribution · Sums of Independent Random Variables · Normal ApproximationVerified
  56. SOA Exam P Sample Question #330Multivariate Random Variables · Variance of Linear Combinations · Independence · CovarianceVerified
  57. SOA Exam P Sample Question #338Multivariate Random Variables · Joint Probability Mass Functions · Conditional Expectation · Discrete Random VariablesVerified
  58. SOA Exam P Sample Question #345Multivariate Random Variables · Normal Distribution · Linear Combinations of Independent Random Variables · Variance of Independent SumsVerified
  59. SOA Exam P Sample Question #373Multivariate Random Variables · Marginal Distributions · VarianceVerified
  60. SOA Exam P Sample Question #382Multivariate Random Variables · Hypergeometric Distribution · Conditional Distributions · Conditional VarianceVerified
  61. SOA Exam P Sample Question #400Multivariate Random Variables · Joint Distributions · Joint Probabilities · Complement RuleVerified
  62. SOA Exam P Sample Question #404Multivariate Random Variables · Marginal Distributions · VarianceVerified
  63. SOA Exam P Sample Question #410Multivariate Random Variables · Joint Cumulative Distribution Functions · Compound Distributions · ConditioningVerified
  64. SOA Exam P Sample Question #411Multivariate Random Variables · Joint Probability Mass Functions · Marginal Distributions · VarianceVerified
  65. SOA Exam P Sample Question #416Multivariate Random Variables · Poisson Distribution · Normal Approximation · Sums of Independent Random VariablesVerified
  66. SOA Exam P Sample Question #431Multivariate Random Variables · Normal Distribution · Variance of Independent SumsVerified
  67. SOA Exam P Sample Question #433Multivariate Random Variables · Normal Distribution · Linear Combinations · Variance of Independent SumsVerified
  68. SOA Exam P Sample Question #439Multivariate Random Variables · Normal Distribution · Sums of Independent Random VariablesVerified
  69. SOA Exam P Sample Question #444Multivariate Random Variables · Joint Distributions · Expectation of a FunctionVerified
  70. SOA Exam P Sample Question #446Multivariate Random Variables · Joint Distributions · Marginal Distributions · Expectation and VarianceVerified
  71. SOA Exam P Sample Question #448Multivariate Random Variables · Order Statistics · Deductibles · Distribution FunctionsVerified
  72. SOA Exam P Sample Question #449Multivariate Random Variables · Joint Distributions · Joint ProbabilitiesVerified
  73. SOA Exam P Sample Question #450Multivariate Random Variables · Poisson Distribution · Linear Combinations · Coefficient of VariationVerified
  74. SOA Exam P Sample Question #451Multivariate Random Variables · Exponential Distribution · Order StatisticsVerified
  75. SOA Exam P Sample Question #458Multivariate Random Variables · Linear Combinations of Random Variables · Variance Minimization · IndependenceVerified
  76. SOA Exam P Sample Question #459Multivariate Random Variables · Central Limit Theorem · Sampling DistributionsVerified
  77. SOA Exam P Sample Question #466Multivariate Random Variables · Bernoulli Distribution · Covariance and Correlation · Joint ProbabilitiesVerified
  78. SOA Exam P Sample Question #467Multivariate Random Variables · Joint Distributions · Expectation of a FunctionVerified
  79. SOA Exam P Sample Question #471Multivariate Random Variables · Joint Distributions · Marginal Distributions · VarianceVerified
  80. SOA Exam P Sample Question #474Multivariate Random Variables · Exponential Distribution · Sums of Independent Random Variables · Variance and Standard DeviationVerified
  81. SOA Exam P Sample Question #476Multivariate Random Variables · Joint Distributions · Conditional ProbabilityVerified
  82. SOA Exam P Sample Question #478Multivariate Random Variables · Sums of Independent Random Variables · Variance and Standard DeviationVerified
  83. SOA Exam P Sample Question #483Multivariate Random Variables · Independence and Unions · Binomial DistributionVerified
  84. SOA Exam P Sample Question #509Multivariate Random Variables · Normal Distribution · Linear Combinations of Independent Random Variables · StandardizationVerified
  85. SOA Exam P Sample Question #510Multivariate Random Variables · Joint Distributions · Transformations · Expected ValueVerified
  86. SOA Exam P Sample Question #514Multivariate Random Variables · Joint Distributions · Expected ValueVerified
  87. SOA Exam P Sample Question #517Multivariate Random Variables · Joint Distributions · Marginal Distributions · VarianceVerified
  88. SOA Exam P Sample Question #519Multivariate Random Variables · Exponential Distribution · Linear Combinations of Independent Random Variables · Variance and Standard DeviationVerified
  89. SOA Exam P Sample Question #520Multivariate Random Variables · Conditional Distributions · Bernoulli Distribution · Conditional VarianceVerified
  90. SOA Exam P Sample Question #521Multivariate Random Variables · Normal Distribution · Linear Combinations of Independent Random Variables · IndependenceVerified
  91. SOA Exam P Sample Question #525Multivariate Random Variables · Joint Distributions · Conditional Distributions · VarianceVerified
  92. SOA Exam P Sample Question #526Multivariate Random Variables · Central Limit Theorem · Sums of Independent Random VariablesVerified
  93. SOA Exam P Sample Question #527Multivariate Random Variables · Covariance and Correlation · Linear CombinationsVerified
  94. SOA Exam P Sample Question #535Multivariate Random Variables · Linear Combinations of Independent Random Variables · VarianceVerified
  95. SOA Exam P Sample Question #539Multivariate Random Variables · Normal Distribution · Linear CombinationsVerified
  96. SOA Exam P Sample Question #542Multivariate Random Variables · Sums of Independent Random Variables · Variance and Standard DeviationVerified
  97. SOA Exam P Sample Question #555Multivariate Random Variables · Normal Distribution · Sums of Independent Random VariablesVerified
  98. SOA Exam P Sample Question #569Multivariate Random Variables · Sums of Independent Random Variables · Binomial DistributionVerified
  99. SOA Exam P Sample Question #574Multivariate Random Variables · Joint Distributions · Marginal Distributions · VarianceVerified
  100. SOA Exam P Sample Question #575Multivariate Random Variables · Normal Distribution · Sums of Independent Random VariablesVerified
  101. SOA Exam P Sample Question #577Multivariate Random Variables · Sums of Independent Random Variables · Coefficient of VariationVerified
  102. SOA Exam P Sample Question #578Multivariate Random Variables · Normal Distribution · Sums of Independent Random VariablesVerified
  103. SOA Exam P Sample Question #580Multivariate Random Variables · Linear Combinations of Independent Random Variables · VarianceVerified
  104. SOA Exam P Sample Question #581Multivariate Random Variables · Joint Distributions · Covariance and CorrelationVerified
  105. SOA Exam P Sample Question #582Multivariate Random Variables · Variance and Standard Deviation · Sums of Independent Random VariablesVerified
  106. SOA Exam P Sample Question #587Multivariate Random Variables · Joint Distributions · Marginal Distributions · VarianceVerified
  107. SOA Exam P Sample Question #606Multivariate Random Variables · Joint Distributions · Conditional Distributions · VarianceVerified
  108. SOA Exam P Sample Question #614Multivariate Random Variables · Normal Distribution · Linear CombinationsVerified
  109. SOA Exam P Sample Question #617Multivariate Random Variables · Conditional Distributions · VarianceVerified
  110. SOA Exam P Sample Question #624Multivariate Random Variables · Normal Distribution · Linear Combinations of Independent Random Variables · IndependenceVerified
  111. SOA Exam P Sample Question #627Multivariate Random Variables · Conditional Distributions · Conditional ExpectationVerified
  112. SOA Exam P Sample Question #631Multivariate Random Variables · Conditional Distributions · Bernoulli Distribution · Conditional VarianceVerified
  113. SOA Exam P Sample Question #633Multivariate Random Variables · Poisson Distribution · Conditional DistributionsVerified
  114. SOA Exam P Sample Question #634Multivariate Random Variables · Sums of Independent Random Variables · Coefficient of VariationVerified
  115. SOA Exam P Sample Question #635Multivariate Random Variables · Exponential Distribution · Means and VariancesVerified
  116. SOA Exam P Sample Question #636Multivariate Random Variables · Central Limit Theorem · Sums of Independent Random VariablesVerified
  117. SOA Exam P Sample Question #637Multivariate Random Variables · Central Limit Theorem · Sums of Independent Random VariablesVerified
  118. SOA Exam P Sample Question #638Multivariate Random Variables · Conditional Distributions · VarianceVerified
  119. SOA Exam P Sample Question #650Multivariate Random Variables · Multinomial Distribution · Linearity of ExpectationVerified
  120. SOA Exam P Sample Question #651Multivariate Random Variables · Marginal Distributions · Expected LossVerified
  121. SOA Exam P Sample Question #652Multivariate Random Variables · Marginal Distributions · MomentsVerified
  122. SOA Exam P Sample Question #656Multivariate Random Variables · Binomial and Multinomial Distributions · Complement RuleVerified
  123. SOA Exam P Sample Question #662Multivariate Random Variables · Discrete Random Variables · Sums of Independent Random Variables · ConvolutionVerified
  124. SOA Exam P Sample Question #671Multivariate Random Variables · Linear Combinations of Independent Random Variables · Normal Distribution · IndependenceVerified
  125. SOA Exam P Sample Question #673Multivariate Random Variables · Covariance and Correlation · Linear CombinationsVerified
  126. SOA Exam P Sample Question #676Multivariate Random Variables · Poisson Distribution · Sums of Independent Random VariablesVerified
  127. SOA Exam P Sample Question #681Multivariate Random Variables · Expectation and Variance · Sums of Independent Random VariablesVerified
  128. SOA Exam P Sample Question #686Multivariate Random Variables · Expected Value · Linearity of Expectation · Insurance Loss ModelingVerified
  129. SOA Exam P Sample Question #687Multivariate Random Variables · Poisson Distribution · Normal Approximation · Sums of Independent Random VariablesVerified
  130. SOA Exam P Sample Question #689Multivariate Random Variables · Conditional Distributions · Law of Total Expectation · Discrete Uniform DistributionVerified
  131. SOA Exam P Sample Question #690Multivariate Random Variables · Discrete Random Variables · Sums of Independent Random Variables · Probability Generating FunctionsVerified
  132. SOA Exam P Sample Question #692Multivariate Random Variables · Central Limit Theorem · Normal ApproximationVerified
  133. SOA Exam P Sample Question #702Multivariate Random Variables · Joint Distributions · Covariance and CorrelationVerified
  134. SOA Exam P Sample Question #716Multivariate Random Variables · Conditional Distributions · Law of Total Expectation · Expected ValueVerified
  135. SOA Exam P Sample Question #717Multivariate Random Variables · Conditional Distributions · VarianceVerified
  136. SOA Exam P Sample Question #721Multivariate Random Variables · Linear Combinations · Normal Distribution · PercentilesVerified
  137. SOA Exam P Sample Question #724Multivariate Random Variables · Linear Combinations of Independent Random Variables · Normal Distribution · PercentilesVerified
  138. SOA Exam P Sample Question #726Multivariate Random Variables · Joint Distributions · Conditional Distributions · Expectation and VarianceVerified
  139. SOA Exam P Sample Question #735Multivariate Random Variables · Joint Distributions · Expected ValueVerified

Drill one method at a time

Techniques that keep appearing in Multivariate Random Variables

Each guide collects the questions above that turn on the same method, with the formulas and the errors that cost marks.

57 worked solutions

Exam P Variance problems

Second moments, transformations, sums, covariance terms, and conditional variance decompositions.

55 worked solutions

Exam P Independence problems

Event independence, factorization, independent trials, and the consequences independence does—and does not—justify.

46 worked solutions

Exam P Expected Value problems

Discrete sums, continuous integrals, functions of loss, linearity, conditioning, and tail-based expectation.

Study the whole topic offline

Every Exam P sample question in one searchable PDF

The 139 pages above stay free. The manual collects the complete Exam P sample set in syllabus order, with the same working, the wrong-choice notes, and one new practice problem per question.