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Erand Weibull Series Parallel SurvivalErand Weibull Hazard Median Mean

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Random variables and engineering distribution modelsEngineering Probability and Random Processes · Mixed review

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Erand Geometric Stopped Exponential SumErand Laplace Total Variance Audit

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Random variables and engineering distribution modelsEngineering Probability and Random Processes · Mixed review

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Erand Shared Environment Joint MomentsErand Conditional VS Unconditional Independence

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Joint laws, transformations, and conditional expectationEngineering Probability and Random Processes · Mixed review

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Erand Lognormal Pair Moments CorrelationErand Gaussian Linear Combo Independence

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Joint laws, transformations, and conditional expectationEngineering Probability and Random Processes · Mixed review

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Erand Sampling Without Replacement MomentsErand Hypergeometric Sum Mean Variance

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Joint laws, transformations, and conditional expectationEngineering Probability and Random Processes · Mixed review

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Erand Iid Increment Min CovarianceErand Walk Not Wss White Increments

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Random processes and stationarityEngineering Probability and Random Processes · Mixed review

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Erand Telegraph Conditional MeanErand Telegraph Lorentzian Psd

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Random processes and stationarityEngineering Probability and Random Processes · Mixed review

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Erand Wss Sample Mean VarianceErand Effective Sample Count

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Correlation, power spectra, and linear systemsEngineering Probability and Random Processes · Mixed review

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Erand Cross Spectrum CoherenceErand Bandlimited Wiener Khinchin

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Correlation, power spectra, and linear systemsEngineering Probability and Random Processes · Mixed review

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Erand Lag One Bochner ValidityErand MA One Realization

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Correlation, power spectra, and linear systemsEngineering Probability and Random Processes · Mixed review

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ERAND:010527Urgent and routine thinnings of a rate-6 stream are independent Poisson of rates 2 and 4; a half-hour window has P(U=2,R=1)=e^{-3} and P(U=2|N=3)=2/9

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Erand Poisson Independent ThinningErand Binomial Coloring Given Count

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Poisson processes and Markov chainsEngineering Probability and Random Processes · Mixed review

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ERAND:010528The NHPP of intensity 2t has first-arrival density 2t e^{-t^2}; P[N(2)-N(1)=3]=(9/2)e^{-3} and E[first time | N(2)=2]=16/15

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Erand Nhpp Mean Function First ArrivalErand Nhpp Conditional Order Statistics

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Poisson processes and Markov chainsEngineering Probability and Random Processes · Mixed review

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ERAND:010529The repair chain with up-to-down rate 1 and down-to-up rate 3 has stationary law (3/4,1/4) and explicit e^{-4t} transitions

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Erand Two State Ctmc Stationary LawErand Two State Ctmc Transition Semigroup

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Poisson processes and Markov chainsEngineering Probability and Random Processes · Mixed review

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ERAND:010630The BLUE of theta from Y=(theta+N1,theta+N2) with covariance [[4,1],[1,2]] uses weights (1/4,3/4), has variance 7/4, and reads 13 at (10,14)

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Erand Blue Two Observation WeightsErand Blue Minimum Variance Evaluate

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Estimation, detection, and noise modelsEngineering Probability and Random Processes · Mixed review

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ERAND:010631The exact 2-bit midrise quantizer of Unif[-1,1) has error variance 1/48, SQNR 16 (about 12.04 dB), and error independent of Q but not of X

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Erand Midrise Uniform Error MomentsErand Sqnr And Error Independence

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Estimation, detection, and noise modelsEngineering Probability and Random Processes · Mixed review

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Erand Weibull Series Parallel SurvivalErand Weibull Hazard Median Mean

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Random variables and engineering distribution modelsEngineering Probability and Random Processes · Mixed review

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Erand Geometric Stopped Exponential SumErand Laplace Total Variance Audit

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Random variables and engineering distribution modelsEngineering Probability and Random Processes · Mixed review

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Erand Shared Environment Joint MomentsErand Conditional VS Unconditional Independence

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Joint laws, transformations, and conditional expectationEngineering Probability and Random Processes · Mixed review

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Erand Lognormal Pair Moments CorrelationErand Gaussian Linear Combo Independence

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Joint laws, transformations, and conditional expectationEngineering Probability and Random Processes · Mixed review

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Erand Sampling Without Replacement MomentsErand Hypergeometric Sum Mean Variance

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Joint laws, transformations, and conditional expectationEngineering Probability and Random Processes · Mixed review

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Erand Iid Increment Min CovarianceErand Walk Not Wss White Increments

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Random processes and stationarityEngineering Probability and Random Processes · Mixed review

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Erand Telegraph Conditional MeanErand Telegraph Lorentzian Psd

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Random processes and stationarityEngineering Probability and Random Processes · Mixed review

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Erand Wss Sample Mean VarianceErand Effective Sample Count

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Correlation, power spectra, and linear systemsEngineering Probability and Random Processes · Mixed review

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Erand Cross Spectrum CoherenceErand Bandlimited Wiener Khinchin

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Correlation, power spectra, and linear systemsEngineering Probability and Random Processes · Mixed review

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Erand Lag One Bochner ValidityErand MA One Realization

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Correlation, power spectra, and linear systemsEngineering Probability and Random Processes · Mixed review

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ERAND:010527Urgent and routine thinnings of a rate-6 stream are independent Poisson of rates 2 and 4; a half-hour window has P(U=2,R=1)=e^{-3} and P(U=2|N=3)=2/9

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Erand Poisson Independent ThinningErand Binomial Coloring Given Count

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Poisson processes and Markov chainsEngineering Probability and Random Processes · Mixed review

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ERAND:010528The NHPP of intensity 2t has first-arrival density 2t e^{-t^2}; P[N(2)-N(1)=3]=(9/2)e^{-3} and E[first time | N(2)=2]=16/15

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Erand Nhpp Mean Function First ArrivalErand Nhpp Conditional Order Statistics

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Poisson processes and Markov chainsEngineering Probability and Random Processes · Mixed review

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ERAND:010529The repair chain with up-to-down rate 1 and down-to-up rate 3 has stationary law (3/4,1/4) and explicit e^{-4t} transitions

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Erand Two State Ctmc Stationary LawErand Two State Ctmc Transition Semigroup

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Poisson processes and Markov chainsEngineering Probability and Random Processes · Mixed review

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ERAND:010630The BLUE of theta from Y=(theta+N1,theta+N2) with covariance [[4,1],[1,2]] uses weights (1/4,3/4), has variance 7/4, and reads 13 at (10,14)

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Erand Blue Two Observation WeightsErand Blue Minimum Variance Evaluate

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Estimation, detection, and noise modelsEngineering Probability and Random Processes · Mixed review

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ERAND:010631The exact 2-bit midrise quantizer of Unif[-1,1) has error variance 1/48, SQNR 16 (about 12.04 dB), and error independent of Q but not of X

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Erand Midrise Uniform Error MomentsErand Sqnr And Error Independence

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Estimation, detection, and noise modelsEngineering Probability and Random Processes · Mixed review

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Problem results

ERAND:010527Urgent and routine thinnings of a rate-6 stream are independent Poisson of rates 2 and 4; a half-hour window has P(U=2,R=1)=e^{-3} and P(U=2|N=3)=2/9

ERAND:010528The NHPP of intensity 2t has first-arrival density 2t e^{-t^2}; P[N(2)-N(1)=3]=(9/2)e^{-3} and E[first time | N(2)=2]=16/15

ERAND:010529The repair chain with up-to-down rate 1 and down-to-up rate 3 has stationary law (3/4,1/4) and explicit e^{-4t} transitions

ERAND:010630The BLUE of theta from Y=(theta+N1,theta+N2) with covariance [[4,1],[1,2]] uses weights (1/4,3/4), has variance 7/4, and reads 13 at (10,14)

ERAND:010631The exact 2-bit midrise quantizer of Unif[-1,1) has error variance 1/48, SQNR 16 (about 12.04 dB), and error independent of Q but not of X

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Problem results

ERAND:010527Urgent and routine thinnings of a rate-6 stream are independent Poisson of rates 2 and 4; a half-hour window has P(U=2,R=1)=e^{-3} and P(U=2|N=3)=2/9

ERAND:010528The NHPP of intensity 2t has first-arrival density 2t e^{-t^2}; P[N(2)-N(1)=3]=(9/2)e^{-3} and E[first time | N(2)=2]=16/15

ERAND:010529The repair chain with up-to-down rate 1 and down-to-up rate 3 has stationary law (3/4,1/4) and explicit e^{-4t} transitions

ERAND:010630The BLUE of theta from Y=(theta+N1,theta+N2) with covariance [[4,1],[1,2]] uses weights (1/4,3/4), has variance 7/4, and reads 13 at (10,14)

ERAND:010631The exact 2-bit midrise quantizer of Unif[-1,1) has error variance 1/48, SQNR 16 (about 12.04 dB), and error independent of Q but not of X

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Problem results

ERAND:010117Three iid Weibull lifetimes with S(t)=exp[-(t/10)^2] have series survival exp[-3(t/10)^2], hazard 3t/50, median 10 sqrt(ln 2 / 3), and mean 5 sqrt(pi/3)

ERAND:010118A Geom(1/2)-stopped sum of rate-2 exponentials has Laplace transform 1/(s+1), is Exp(1), and has mean and variance 1

ERAND:010219Shared-environment binary failures have marginals 1/5, joint 7/100, covariance 3/100, correlation 3/16, and P(Y=1|X=1)=7/20

ERAND:010220The lognormal pair from a standard bivariate Gaussian of correlation 1/2 has means sqrt(e), variances e(e-1), covariance e^{3/2}-e, and independent log-sum and log-ratio

ERAND:010221Two ordered draws without replacement from 2 defective and 3 good items have EX_i=2/5, Cov=-3/50, Corr=-1/4, and sample-mean variance 9/100

ERAND:010322The iid sum walk has Cov(Xm,Xn)=min(m,n), Gram [[1,1,1],[1,2,2],[1,2,3]], and Corr(X2,X4)=1/sqrt(2), but is not WSS

ERAND:010323The rate-3/2 telegraph has conditional mean e^{-3|tau|}X(t), R=e^{-3|tau|}, PSD 6/(omega^2+9), and same-sign probability (1+e^{-3T})/2

ERAND:010424The four-sample mean of R[k]=(1/2)^|k| has variance 33/64, not the independent value 1/4, so n_eff=64/33

ERAND:010425X=S+N1 and Y=2S+N2 have spectra 2,8,2 and MSC 1/4; Wiener-Khinchin then gives variances 2/pi, 8/pi, covariance 2/pi, correlation 1/2

ERAND:010426Ra with R[0]=1 and R[+/-1]=a is a valid covariance iff |a|<=1/2; a=3/5 has S(pi)=-1/5; a=2/5 is MA(1) with Var W=4/5

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Problem results

ERAND:010117Three iid Weibull lifetimes with S(t)=exp[-(t/10)^2] have series survival exp[-3(t/10)^2], hazard 3t/50, median 10 sqrt(ln 2 / 3), and mean 5 sqrt(pi/3)

ERAND:010118A Geom(1/2)-stopped sum of rate-2 exponentials has Laplace transform 1/(s+1), is Exp(1), and has mean and variance 1

ERAND:010219Shared-environment binary failures have marginals 1/5, joint 7/100, covariance 3/100, correlation 3/16, and P(Y=1|X=1)=7/20

ERAND:010220The lognormal pair from a standard bivariate Gaussian of correlation 1/2 has means sqrt(e), variances e(e-1), covariance e^{3/2}-e, and independent log-sum and log-ratio

ERAND:010221Two ordered draws without replacement from 2 defective and 3 good items have EX_i=2/5, Cov=-3/50, Corr=-1/4, and sample-mean variance 9/100

ERAND:010322The iid sum walk has Cov(Xm,Xn)=min(m,n), Gram [[1,1,1],[1,2,2],[1,2,3]], and Corr(X2,X4)=1/sqrt(2), but is not WSS

ERAND:010323The rate-3/2 telegraph has conditional mean e^{-3|tau|}X(t), R=e^{-3|tau|}, PSD 6/(omega^2+9), and same-sign probability (1+e^{-3T})/2

ERAND:010424The four-sample mean of R[k]=(1/2)^|k| has variance 33/64, not the independent value 1/4, so n_eff=64/33

ERAND:010425X=S+N1 and Y=2S+N2 have spectra 2,8,2 and MSC 1/4; Wiener-Khinchin then gives variances 2/pi, 8/pi, covariance 2/pi, correlation 1/2

ERAND:010426Ra with R[0]=1 and R[+/-1]=a is a valid covariance iff |a|<=1/2; a=3/5 has S(pi)=-1/5; a=2/5 is MA(1) with Var W=4/5