Mathematical Statistics 2015
60 questions, booklet MS2015.
Section A · Section A · Q1–30
Practise this section →- 1UMVUE of the reciprocal of an exponential rateEstimationUMVUEMedium
- 2Correlation between two overlapping sums of independent normalsMultivariate DistributionsCorrelation and covarianceEasy
- 3Power of a test on a single uniform observationTesting of HypothesesPower functionMedium
- 4Moment generating function from a pmf proportional to binomial coefficientsUnivariate DistributionsMoment generating functionsEasy
- 5Conditional probability of the complement given a unionDescriptive Statistics and ProbabilityProbabilityEasy
- 6Most powerful test between two gamma shapesTesting of HypothesesMost powerful testsMedium
- 7Transformation that is not linearMatrices and DeterminantsLinear transformationsEasy
- 8Subsequences of a monotone bounded sequenceSequences and SeriesConvergence of sequencesEasy
- 9Injectivity and surjectivity of a cubic polynomialCalculusDifferential calculus of one variableEasy
- 10Inequality between an exponential and a lineCalculusDifferential calculus of one variableEasy
- 11Cramer-Rao bound for a function of a Poisson parameterEstimationCramer-Rao lower boundHard
- 12Comparison of two independent uniform variables on overlapping rangesDescriptive Statistics and ProbabilityProbabilityMedium
- 13Probability that exactly one box has exactly one defective componentDescriptive Statistics and ProbabilityProbabilityMedium
- 14Confidence interval consistent with rejection of the nullEstimationConfidence intervalsHard
- 15MLE of the exponent of a power densityEstimationMaximum likelihood estimationMedium
- 16Consistent estimator for the scale of a quartic-type densityEstimationConsistency of estimatorsMedium
- 17Normalising constant and shift of a Gaussian-type densityUnivariate DistributionsNormal distributionMedium
- 18Type I and Type II errors of a threshold test for an exponential rateTesting of HypothesesType I and Type II errorsMedium
- 19Variance of a Poisson variable with an exponential random meanMultivariate DistributionsConditional expectationHard
- 20Probability that a biscuit has no cashew nutsLimit TheoremsPoisson approximationMedium
- 21MLE of a common variance factorEstimationMaximum likelihood estimationMedium
- 22Distribution of twice the negative log sum of scaled absolute uniformsSampling DistributionsGamma and chi-squareHard
- 23Monotonicity of the absolute value from a sign conditionCalculusDifferential calculus of one variableMedium
- 24Matrix equation satisfied by a two by two matrixMatrices and DeterminantsCayley-Hamilton theoremMedium
- 25Consequence of a zero determinantMatrices and DeterminantsDeterminantsEasy
- 26Number of roots of a logarithmic functionCalculusDifferential calculus of one variableMedium
- 27Number of values of the variable making a matrix singularMatrices and DeterminantsDeterminantsMedium
- 28Derivative of a double integral over a squareCalculusDifferential calculus of two variablesHard
- 29Solution set of a consistent linear system with given rankMatrices and DeterminantsSystems of linear equationsMedium
- 30Limit of n times an integral with a power weightLimit TheoremsConvergence in distributionHard
Section B · Section B · Q31–40
Practise this section →- 31Statements that are necessarily false for series of positive termsSequences and SeriesConvergence of seriesHard
- 32Solutions of a linear first-order differential equationDifferential EquationsFirst order linear equationsMedium
- 33Zeros of a continuous function from three valuesCalculusDifferential calculus of one variableMedium
- 34Even and odd functions from a composite definitionCalculusDifferential calculus of one variableMedium
- 35Matrices that can be a variance-covariance matrixMultivariate DistributionsCorrelation and covarianceMedium
- 36Sufficient statistics for the mean of a normal distributionEstimationSufficient statisticsMedium
- 37Unbiased and consistent estimators of theta squaredEstimationUnbiased estimatorsHard
- 38Expressions that equal the common varianceDescriptive Statistics and ProbabilityExpected valueMedium
- 39Limit theorems for i.i.d. variables with given mean and varianceLimit TheoremsCentral Limit Theorem (i.i.d. and finite variance case)Medium
- 40Properties of estimators in the normal model with both parameters unknownEstimationMaximum likelihood estimationHard
Section C · Section C · Q41–60
Practise this section →- 41Mean of the minimum of two independent exponentialsUnivariate DistributionsExponential distributionEasy
- 42Conditional probability for a geometric random variableUnivariate DistributionsGeometric distributionMedium
- 43MLE of a Bernoulli parameter restricted to an intervalEstimationMaximum likelihood estimationMedium
- 44Expected absolute value of a Laplace variableUnivariate DistributionsMathematical expectation and momentsMedium
- 45Smallest size of a parallel system with a reliability targetDescriptive Statistics and ProbabilityProbabilityMedium
- 46Location of the interval of fixed length with maximum normal probabilityUnivariate DistributionsNormal distributionMedium
- 47Volume of a solid of revolution of a lineCalculusIntegral calculusEasy
- 48Integral of the floor function over an intervalCalculusIntegral calculusEasy
- 49Number of real solutions of a cubic equationCalculusDifferential calculus of one variableMedium
- 50Second derivative of a function constant along a sequenceCalculusDifferential calculus of one variableHard
- 51Mean of a variable whose conditional law is uniform on zero to yMultivariate DistributionsConditional expectationMedium
- 52Probability from a density with a flat part and a power tailUnivariate DistributionsContinuous distributionsMedium
- 53Limit of a sum of uniforms with an n to the three quarters thresholdLimit TheoremsCentral Limit Theorem (i.i.d. and finite variance case)Hard
- 54Least squares prediction from sums of squares and productsEstimationSimple linear regressionMedium
- 55Conditional probability of an atom from a mixed distribution functionUnivariate DistributionsDistribution functionsMedium
- 56Integral of the distribution function of a symmetric variableUnivariate DistributionsSymmetric distributionsHard
- 57Length of an arc of a semicircleCalculusIntegral calculusMedium
- 58Parameter value for which a linear system has no solutionMatrices and DeterminantsSystems of linear equationsHard
- 59Limit at minus infinity of a solution of a linear ODEDifferential EquationsGeneral theory of linear ODEsMedium
- 60Area of a region bounded by a line and a curve in the first quadrantCalculusIntegral calculusMedium