Distribution Theory
55 questions across 3 topics. Also examined in CNET, ISS, GATE ST, GATE DA, JAM MS.
Standard Discrete and Continuous Univariate Distributions
- 127Mean and variance of a mixture distributionMSET June 2025Standard Discrete and Continuous Univariate DistributionsHard
- 138Which families are one-parameter exponential familiesMSET June 2025Standard Discrete and Continuous Univariate DistributionsHard
- 101Exponential distribution — what order statistics do (and don't) inheritMSET April 2024Standard Discrete and Continuous Univariate DistributionsHard
- 127Binomial splitting — the hypergeometric conditional lawMSET April 2024Standard Discrete and Continuous Univariate DistributionsHard
- 128Order statistics of Uniform(0,theta) — a theta-free conditional lawMSET April 2024Standard Discrete and Continuous Univariate DistributionsHard
- 102Normal distribution facts — mean deviation coincides at a symmetric centreMSET March 2023Standard Discrete and Continuous Univariate DistributionsMedium
- 115Extreme order statistics converge to a point, not to a rescaled limit lawMSET March 2023Standard Discrete and Continuous Univariate DistributionsHard
- 123Waiting past two weeks with an exponential lifetimeMSET March 2023Standard Discrete and Continuous Univariate DistributionsEasy
- 128Multinomial marginals are binomial — but never independent of each otherMSET March 2023Standard Discrete and Continuous Univariate DistributionsHard
- 8A basic hypergeometric probabilityMSET September 2021Standard Discrete and Continuous Univariate DistributionsEasy
- 103Normal tail probabilities — comparing the mean to a point above itMSET September 2021Standard Discrete and Continuous Univariate DistributionsEasy
- 109Density of the sample range of iid exponentialsMSET September 2021Standard Discrete and Continuous Univariate DistributionsMedium
- 110Recognising a distribution from its moment generating functionMSET September 2021Standard Discrete and Continuous Univariate DistributionsEasy
- 152Asymptotic normality of bar X/S for exponential data, via the delta methodMSET September 2021Standard Discrete and Continuous Univariate DistributionsHard
- 8Recognising the MGF of a Poisson distributionMSET December 2020Standard Discrete and Continuous Univariate DistributionsEasy
- 103Symmetric tail probabilities around the mean need no explicit sigmaMSET December 2020Standard Discrete and Continuous Univariate DistributionsEasy
- 111Constant hazard rate characterizes the exponential distributionMSET December 2020Standard Discrete and Continuous Univariate DistributionsMedium
- 112A linear combination of iid normal vectors by row-vector weights is scalar (univariate) normalMSET December 2020Standard Discrete and Continuous Univariate DistributionsEasy
- 7A binomial probability with a 2:1 biased coinMSET June 2019Standard Discrete and Continuous Univariate DistributionsEasy
- 109The minimum of n iid exponentials is exponential with mean theta/nMSET June 2019Standard Discrete and Continuous Univariate DistributionsEasy
- 144Poisson: binomial given the total, shifted Poisson given one summandMSET June 2019Standard Discrete and Continuous Univariate DistributionsEasy
- 10A discrete uniform on 2n+1 points: Y=4 means X=pm2MSET January 2018 Paper IIStandard Discrete and Continuous Univariate DistributionsEasy
- 13Normal probabilities by standardising: P(B)<P(A)<P(C)MSET January 2018 Paper IIStandard Discrete and Continuous Univariate DistributionsEasy
- 60Counting failures instead of successes: n-Xsim B(n,1-p)MSET January 2018 Paper IIStandard Discrete and Continuous Univariate DistributionsEasy
- 61Gamma variables with a common scale: X/(X+Y)sim Beta(alpha,beta)MSET January 2018 Paper IIStandard Discrete and Continuous Univariate DistributionsMedium
- 62Given the total, X1/Tsim Beta(1,n-1)MSET January 2018 Paper IIStandard Discrete and Continuous Univariate DistributionsMedium
- 142F(1.5)=P(Xle1)=(1+lambda)e^-lambdaMSET January 2018 Paper IIIStandard Discrete and Continuous Univariate DistributionsEasy
- 58The square of a standard normal is chi^21MSET April 2017 Paper IIStandard Discrete and Continuous Univariate DistributionsEasy
- 592X1+X2 has mean 3lambda but variance 5lambda: not PoissonMSET April 2017 Paper IIStandard Discrete and Continuous Univariate DistributionsEasy
- 60log(1+X) is exponential with rate lambda; two of them add to Gamma(2,lambda)MSET April 2017 Paper IIStandard Discrete and Continuous Univariate DistributionsMedium
- 62-log Xi is exponential with rate theta, so the sum is Gamma(n,theta)MSET April 2017 Paper IIStandard Discrete and Continuous Univariate DistributionsEasy
- 93A multinomial count is binomial: success = category 1MSET April 2017 Paper IIIStandard Discrete and Continuous Univariate DistributionsEasy
Sampling Distributions, Standard Errors, Asymptotic Distributions
- 140Fisher's variance-stabilising z-transformationMSET June 2025Sampling Distributions, Standard Errors, Asymptotic DistributionsHard
- 153Delta method for four transformations — checking each derivativeMSET September 2021Sampling Distributions, Standard Errors, Asymptotic DistributionsHard
- 152Asymptotic normality of an order statistic near the first quartile of a Pareto-type distributionMSET December 2020Sampling Distributions, Standard Errors, Asymptotic DistributionsHard
- 153Delta method applied to logbar X for a Gamma-distributed sampleMSET December 2020Sampling Distributions, Standard Errors, Asymptotic DistributionsMedium
- 155Deriving the Fisher z-transformation from the delta methodMSET December 2020Sampling Distributions, Standard Errors, Asymptotic DistributionsMedium
- 152Dividing by sum Xi^2 rather than its square root leaves a N(0,1/sigma^2) limitMSET June 2019Sampling Distributions, Standard Errors, Asymptotic DistributionsMedium
- 154The ARE of the zero-frequency estimator is lambda/(e^lambda-1), not a listed optionMSET June 2019Sampling Distributions, Standard Errors, Asymptotic DistributionsHard
- 155A vanishing c/sqrt n shift keeps consistency but moves the normal limit to N(c,1)MSET June 2019Sampling Distributions, Standard Errors, Asymptotic DistributionsMedium
- 51A sample mean 1.96 standard errors above mu: odds of 39:1 againstMSET January 2018 Paper IISampling Distributions, Standard Errors, Asymptotic DistributionsEasy
- 59If Xsim F(m,n) then 1/Xsim F(n,m)MSET January 2018 Paper IISampling Distributions, Standard Errors, Asymptotic DistributionsEasy
- 104For the Cauchy, the median is asymptotically normal and the mean is not even consistentMSET January 2018 Paper IIISampling Distributions, Standard Errors, Asymptotic DistributionsEasy
- 105nFn(x)simBin(n,F(x)), so Fn(x) is asymptotically normalMSET January 2018 Paper IIISampling Distributions, Standard Errors, Asymptotic DistributionsEasy
- 106Delta method with g(lambda)=lambda e^-lambdaMSET January 2018 Paper IIISampling Distributions, Standard Errors, Asymptotic DistributionsEasy
- 74Fisher's z: the difference is 1.83 standard errors — significant at 10\%, not at 5\%MSET April 2017 Paper IISampling Distributions, Standard Errors, Asymptotic DistributionsMedium
- 103sqrt n(2bar X-theta)oversetdlongrightarrow N(0,theta^2/3), so the square is theta^2chi^21/3MSET April 2017 Paper IIISampling Distributions, Standard Errors, Asymptotic DistributionsMedium
- 105nbar X^2 is chi^21 when theta=0 (non-central otherwise)MSET April 2017 Paper IIISampling Distributions, Standard Errors, Asymptotic DistributionsMedium
Distribution of Order Statistics and Range
- 126Order statistics — the maximum of two uniforms is Beta(2,1)MSET April 2024Distribution of Order Statistics and RangeEasy
- 113The extreme order statistics are jointly sufficient for a fixed-width uniform location parameterMSET December 2020Distribution of Order Statistics and RangeMedium
- 154At n=2 the sample mean and the midrange are literally the same statisticMSET December 2020Distribution of Order Statistics and RangeEasy
- 98The range of a U(theta,theta+1) sample has mean (n-1)/(n+1)MSET January 2018 Paper IIIDistribution of Order Statistics and RangeEasy
- 9R\'enyi's representation: the maximum of n unit exponentials equals sum Xi/i in lawMSET April 2017 Paper IIDistribution of Order Statistics and RangeHard
- 61P(max=x)=F(x)^m-F(x-1)^m for integer-valued variablesMSET April 2017 Paper IIDistribution of Order Statistics and RangeMedium
- 106X(1)to-theta and X(n)totheta, so the range estimates 2thetaMSET April 2017 Paper IIIDistribution of Order Statistics and RangeMedium