Mathematical Statistics 2017
60 questions, booklet MS2017.
Section A · Section A · Q1–30
Practise this section →- 1Imaginary parts of eigenvalues of a block triangular matrixMatrices and DeterminantsEigenvalues and eigenvectorsEasy
- 2Rank-one system with non-parallel vectorsMatrices and DeterminantsSystems of linear equationsMedium
- 3Limit and series of a sequence with exponent tending to zeroSequences and SeriesConvergence of sequencesMedium
- 4Monotonicity and limit of a linear recurrenceSequences and SeriesRecurrence relationsMedium
- 5Interval probability from a two-piece densityUnivariate DistributionsContinuous distributionsEasy
- 6Tail probability from a binomial mgfUnivariate DistributionsMoment generating functionsMedium
- 7Point probability from an unnormalised pmfDescriptive Statistics and ProbabilityProbabilityEasy
- 8Uniform variable with cosine exceeding sineUnivariate DistributionsUniform distributionMedium
- 9Point probability of a sample meanLimit TheoremsConvergence in distributionEasy
- 10Linear unbiased estimator with smallest varianceEstimationUnbiased estimatorsMedium
- 11Heads in four tosses exceeding a die rollDescriptive Statistics and ProbabilityProbabilityMedium
- 12Chebyshev lower bound for a sum of two beta variablesUnivariate DistributionsChebyshev inequalityMedium
- 13Tail probability of a sum of three geometric variablesUnivariate DistributionsGeometric distributionHard
- 14Mean of a variable from a triangular joint densityMultivariate DistributionsJoint densitiesMedium
- 15Probability that a sum exceeds one halfMultivariate DistributionsJoint densitiesMedium
- 16Distribution of a ratio built from normal samplesSampling DistributionsF distributionMedium
- 17Interval containing the limit of the sample meanLimit TheoremsWeak law of large numbersMedium
- 18Maximum likelihood estimate of the square of a location parameterEstimationMaximum likelihood estimationMedium
- 19Method of moments for a uniform on a symmetric intervalEstimationMethod of momentsMedium
- 20Constant for a confidence interval of a normal varianceEstimationConfidence intervalsMedium
- 21Power of a test for a difference of normal meansTesting of HypothesesPower functionHard
- 22Power of the most powerful test between two uniformsTesting of HypothesesMost powerful testsHard
- 23Limits after reversing a double integralCalculusIntegral calculusMedium
- 24Second derivative of an integral at zeroCalculusDifferential calculus of one variableHard
- 25Local minimum and maximum of a function of two variablesCalculusDifferential calculus of two variablesMedium
- 26Convergence of a series with a parameter by the root testSequences and SeriesTests of convergenceHard
- 27Parameter making a difference of reciprocals finiteCalculusDifferential calculus of one variableMedium
- 28Bounds for an integral with a cubic under the rootCalculusIntegral calculusHard
- 29Matrix with the same eigenvalues as a given productMatrices and DeterminantsProperties of matricesMedium
- 30Arc length up to a vertical tangentCalculusIntegral calculusHard
Section B · Section B · Q31–40
Practise this section →- 31Properties of a Householder reflectionMatrices and DeterminantsEigenvalues and eigenvectorsHard
- 32Convergence of the sum of an increasing and a decreasing sequenceSequences and SeriesConvergence of sequencesMedium
- 33A bijection of the unit interval defined by rationalityCalculusDifferential calculus of one variableHard
- 34Curve below two lines from a bounded derivativeCalculusDifferential calculus of one variableHard
- 35Probability comparisons when P(E)+P(F) is at least oneDescriptive Statistics and ProbabilityProbabilityHard
- 36Probabilities and mean from a step distribution functionUnivariate DistributionsDistribution functionsMedium
- 37Unbiased estimators of a Bernoulli parameterEstimationUnbiased estimatorsMedium
- 38Variances that an unbiased estimator cannot attainEstimationCramer-Rao lower boundHard
- 39Sufficient but not complete statistics for a gamma familyEstimationSufficiency and completenessHard
- 40Values of the sample total at which the UMP test rejectsTesting of HypothesesMost powerful testsMedium
Section C · Section C · Q41–60
Practise this section →- 41Mean of a negative log of a uniform variableUnivariate DistributionsTransformations of random variablesEasy
- 42Probability from a base ten lognormal variableUnivariate DistributionsNormal distributionMedium
- 43Comparing two averages of independent normal variablesSampling DistributionsSampling distributions from normalMedium
- 44Conditional probability from a pmf with squaresMultivariate DistributionsConditional distributionsEasy
- 45Nine times a covariance from a triangular densityMultivariate DistributionsCorrelation and covarianceHard
- 46Constant and degrees of freedom of a two-sample t statisticSampling Distributionst distributionHard
- 47Coefficient combination from a local minimumCalculusDifferential calculus of one variableMedium
- 48Area between two opposite parabolasCalculusIntegral calculusEasy
- 49Determinant of a product of column-replaced identitiesMatrices and DeterminantsDeterminantsMedium
- 50Telescoping sum of a rational seriesSequences and SeriesInfinite series and its convergenceMedium
- 51Sum of absolute components of a unit null vectorMatrices and DeterminantsSystems of linear equationsHard
- 52Number of roots of an integral function on an intervalCalculusIntegral calculusHard
- 53Length of a closed curve formed by a curve and its tangentCalculusIntegral calculusHard
- 54Volume of a solid under a plane over a parabolic regionCalculusIntegral calculusHard
- 55Probability from a piecewise-linear density with unknown middle heightUnivariate DistributionsContinuous distributionsMedium
- 56Tail of the minimum of exponentials beyond mean plus sdUnivariate DistributionsExponential distributionMedium
- 57Conditional mean on a wedge-shaped supportMultivariate DistributionsConditional expectationMedium
- 58Combination of probabilities from conditional informationMultivariate DistributionsConditional distributionsMedium
- 59Variance from a central limit approximationLimit TheoremsCentral Limit Theorem (i.i.d. and finite variance case)Medium
- 60Probability from a gamma law given its mean and varianceUnivariate DistributionsGamma distributionMedium