Mathematical Statistics 2019
60 questions, booklet MS2019.
Section A · Section A · Q1–30
Practise this section →- 1Convergent sequence composed with expSequences and SeriesConvergence of sequencesEasy
- 2Monkey saddle at the originCalculusDifferential calculus of one and two variablesMedium
- 3Derivative of an integral with variable lower limitCalculusIntegral calculusEasy
- 4Linear map given on a basis of R2Matrices and DeterminantsLinear transformationsMedium
- 5Exactly two heads from two biased coinsDescriptive Statistics and ProbabilityProbabilityMedium
- 6Mean of a discrete pmf with unknown constantsDescriptive Statistics and ProbabilityProbabilityMedium
- 7Poisson parameter from a pmf relationUnivariate DistributionsPoisson distributionMedium
- 8Sequence converging to zero in probabilityLimit TheoremsWeak law of large numbersMedium
- 9Unbiased estimator under a normal mixtureEstimationUnbiased estimatorsMedium
- 10LRT after an exponential transformationTesting of HypothesesLikelihood ratio testsMedium
- 11Telescoping sum of a rational seriesSequences and SeriesInfinite series and its convergenceMedium
- 12Limit with harmonic sum and exponential rootsSequences and SeriesConvergence of sequencesMedium
- 13Parameter for which a quartic has no real rootCalculusDifferential calculus of one variableHard
- 14Taylor coefficient of one over cubeSequences and SeriesPower series (of one real variable)Easy
- 15Length of a fractional-power curveCalculusIntegral calculusMedium
- 16Volume of a solid of revolution about x equals 6CalculusIntegral calculusMedium
- 17Singularity of a matrix factored through R2Matrices and DeterminantsRank and nullityMedium
- 18Inverse expressed through Cayley-HamiltonMatrices and DeterminantsCayley-Hamilton theoremMedium
- 19Probability of E without F or GDescriptive Statistics and ProbabilityProbabilityMedium
- 20Statement not true for independent eventsDescriptive Statistics and ProbabilityProbabilityEasy
- 21Chi-square law of twice a squared variableSampling DistributionsGamma and chi-squareMedium
- 22Mean and tail of a logistic variableUnivariate DistributionsMathematical expectation and momentsMedium
- 23Poisson approximation for failing bulbsUnivariate DistributionsPoisson distributionMedium
- 24Variance of a beta distributionUnivariate DistributionsContinuous distributionsHard
- 25Probability that one uniform exceeds the sum of twoMultivariate DistributionsJoint densitiesMedium
- 26Conditional mean given X exceeds YMultivariate DistributionsConditional expectationMedium
- 27MLE of the parameter of N(theta, theta)EstimationMaximum likelihood estimationHard
- 28Sufficient but not complete statisticEstimationConsistency and completenessHard
- 29Most powerful test between two symmetric densitiesTesting of HypothesesMost powerful testsMedium
- 30Minimum sample size for a size constraintTesting of HypothesesType I and Type II errorsMedium
Section B · Section B · Q31–40
Practise this section →- 31Series derived from a convergent positive seriesSequences and SeriesConvergence of seriesHard
- 32Extrema and monotonicity around a pointCalculusDifferential calculus of one variableMedium
- 33Matrix whose every vector is an eigenvectorMatrices and DeterminantsEigenvalues and eigenvectorsMedium
- 34Skew-symmetric matrix of even orderMatrices and DeterminantsSkew-symmetric matricesHard
- 35Interval probabilities from a cdf with jumpsUnivariate DistributionsDistribution functionsMedium
- 36Positive part of a difference of exponentialsUnivariate DistributionsExponential distributionMedium
- 37Independent Poisson pairMultivariate DistributionsJoint distributionsMedium
- 38Sequences converging to a normal with mean zeroLimit TheoremsCentral Limit Theorem (i.i.d. and finite variance case)Medium
- 39Consistent estimators built from the sample minimumEstimationConsistency of estimatorsMedium
- 40Most powerful test for a Weibull-type scaleTesting of HypothesesNeyman-Pearson lemmaMedium
Section C · Section C · Q41–60
Practise this section →- 41Riemann-sum limit with square rootsSequences and SeriesConvergence of sequencesMedium
- 42Function satisfying a Holder condition of exponent 4/3CalculusDifferential calculus of one variableMedium
- 43Double integral by reversing the orderCalculusIntegral calculusMedium
- 44Missing entries of an orthogonal matrixMatrices and DeterminantsOrthogonal matricesMedium
- 45Sum below six given one odd and one even faceDescriptive Statistics and ProbabilityProbabilityEasy
- 46Chebyshev bound from a moment generating functionUnivariate DistributionsChebyshev inequalityMedium
- 47Probability that at least two of three packets passDescriptive Statistics and ProbabilityProbabilityMedium
- 48Expectation in a two-stage coin tossingMultivariate DistributionsConditional expectationMedium
- 49Method of moments for a transformed BernoulliEstimationMethod of momentsMedium
- 50Confidence interval for a difference of meansEstimationConfidence intervalsHard
- 51Second-order limit of the defining sequence of eSequences and SeriesConvergence of sequencesHard
- 52Sum of floor and fractional-part terms near zeroCalculusDifferential calculus of one variableHard
- 53Volume between two planes inside a cylinderCalculusIntegral calculusHard
- 54Determinant of a four by four block matrixMatrices and DeterminantsDeterminantsMedium
- 55Posterior probability of no heart diseaseDescriptive Statistics and ProbabilityBayes theoremMedium
- 56Second moment of a piecewise densityUnivariate DistributionsMathematical expectation and momentsMedium
- 57Variance using the law of total varianceMultivariate DistributionsConditional expectationHard
- 58Expected reciprocal of a chi-square variableSampling DistributionsChi-square and related momentsMedium
- 59Confidence interval based on the sample minimumEstimationConfidence intervalsHard
- 60Size plus power of a test on a Bernoulli sampleTesting of HypothesesPower and Type-II errorMedium