Mathematical Statistics 2018
60 questions, booklet MS2018.
Section A · Section A · Q1–30
Practise this section →- 1Bounds for a recursively defined sequenceSequences and SeriesRecurrence relationsMedium
- 2Limit of a power of one plus two over nSequences and SeriesConvergence of sequencesMedium
- 3Convergence of the maximum and minimum sequencesSequences and SeriesConvergence of sequencesEasy
- 4Matrix equation from the characteristic polynomialMatrices and DeterminantsCayley-Hamilton theoremMedium
- 5Gamma parameters from mean and varianceUnivariate DistributionsGamma distributionEasy
- 6Point masses and tail of a mixed distributionUnivariate DistributionsDistribution functionsMedium
- 7Expected ratio of partial sum to totalMultivariate DistributionsConditional expectationMedium
- 8Maximum likelihood estimate from a categorical sampleEstimationMaximum likelihood estimationMedium
- 9Posterior probability of a coin given a headDescriptive Statistics and ProbabilityBayes theoremMedium
- 10Maximum likelihood estimates in simple regressionEstimationSimple linear regressionMedium
- 11Discontinuities of a function with a floor termCalculusDifferential calculus of one variableHard
- 12Number of intersections of two curvesCalculusDifferential calculus of one variableHard
- 13Minimum of a quartic over a half planeCalculusDifferential calculus of two variablesHard
- 14Sum of outer products of orthonormal vectorsMatrices and DeterminantsOrthogonal matricesMedium
- 15Differentiability of an integral function at breakpointsCalculusIntegral calculusMedium
- 16Value of a combination fixed by two equationsMatrices and DeterminantsSystems of linear equationsMedium
- 17Dimensions of null spaces of a matrixMatrices and DeterminantsRank and nullityMedium
- 18Invertibility of shifts of a skew-symmetric matrixMatrices and DeterminantsSkew-symmetric matricesMedium
- 19Probability of rational values from a mixture of normalsMultivariate DistributionsMoment generating functionsHard
- 20Mean and variance from a product mgfUnivariate DistributionsMoment generating functionsHard
- 21Limit of chi-square tail probabilitiesLimit TheoremsCentral Limit Theorem (i.i.d. and finite variance case)Hard
- 22Expected factorial of a shifted Poisson variableUnivariate DistributionsPoisson distributionMedium
- 23Sign of a cubic in a standard normal variableUnivariate DistributionsNormal distributionMedium
- 24Conditional moments given X for a joint densityMultivariate DistributionsConditional expectationMedium
- 25Conditional pmf from a joint pmfMultivariate DistributionsConditional distributionsEasy
- 26Density of a ratio of absolute standard normalsMultivariate DistributionsOrder statistics and transformationsMedium
- 27Correlation for an ordered exponential pairMultivariate DistributionsCorrelation and covarianceHard
- 28Method of moments for a symmetric discrete uniformEstimationMethod of momentsHard
- 29Most powerful test between two discrete uniformsTesting of HypothesesMost powerful testsHard
- 30Size and power of a given critical regionTesting of HypothesesType I and Type II errorsMedium
Section B · Section B · Q31–40
Practise this section →- 31Monotone bounded sequence from a harmonic blockSequences and SeriesConvergence of sequencesMedium
- 32Series derived from a convergent positive seriesSequences and SeriesConvergence of seriesHard
- 33Divergent increasing recursive sequenceSequences and SeriesRecurrence relationsMedium
- 34Extremum, monotonicity and differentiability at zeroCalculusDifferential calculus of one variableHard
- 35Pairwise and mutual independence on four pointsDescriptive Statistics and ProbabilityProbabilityEasy
- 36Confidence intervals for a shift parameterEstimationConfidence intervalsHard
- 37Consistent estimators of the cube of a uniform endpointEstimationConsistency of estimatorsHard
- 38MLE when the support depends on the parameterEstimationMaximum likelihoodHard
- 39UMVUE of the rate parameter of a gamma familyEstimationUMVUE and completenessHard
- 40Consistent estimators from the extremes of a shifted uniformEstimationConsistency of estimatorsHard
Section C · Section C · Q41–60
Practise this section →- 41Series with a sum of odd numbers over factorialSequences and SeriesInfinite series and its convergenceMedium
- 42Area between two tangent semicirclesCalculusIntegral calculusMedium
- 43Area of the region absolute x plus absolute y at most oneCalculusIntegral calculusEasy
- 44Value of a beta integralUnivariate DistributionsContinuous distributionsEasy
- 45Exactly two sixes given at least one sixDescriptive Statistics and ProbabilityProbabilityMedium
- 46Expectation of a product from a product pmfMultivariate DistributionsJoint distributionsMedium
- 47Conditional probability under conditional independenceDescriptive Statistics and ProbabilityProbabilityMedium
- 48Probability that none of three events occursDescriptive Statistics and ProbabilityProbabilityMedium
- 49Limit in probability of the sample mean of a mixtureLimit TheoremsWeak law of large numbersMedium
- 50MLE of an exponential mean restricted to integersEstimationMaximum likelihoodHard
- 51Limit of scaled differences of a differentiable functionCalculusDifferential calculus of one variableMedium
- 52Rank of a matrix built from four row vectorsMatrices and DeterminantsRank of a matrixHard
- 53Limit of a power with a slowly growing base termCalculusDifferential calculus of one variableHard
- 54Double integral by reversing the orderCalculusIntegral calculusMedium
- 55Probability from a piecewise density with unknown powerUnivariate DistributionsContinuous distributionsHard
- 56Probability from a joint moment generating functionMultivariate DistributionsMoment generating functionsMedium
- 57Probability that four geometric variables sum to sixUnivariate DistributionsGeometric distributionMedium
- 58Expected value of the larger of a variable and fiveDescriptive Statistics and ProbabilityExpected valueEasy
- 59Size plus power of a test for a uniform on moving supportTesting of HypothesesType I and Type II errorsMedium
- 60At least one six in four rollsDescriptive Statistics and ProbabilityProbabilityEasy