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MTP:010117Mixed law (1/4)δ0+(3/4)Exp(rate 2) has F(0)=1/4, density (3/2)e^{-2x} on (0,∞), E=3/8, Var=15/64, median (1/2)log(3/2)

Measure Theoretic Probability Mixed Lebesgue CdfMeasure Theoretic Probability Mixed Moments Laplace

Probability measures, random variables, and induced lawsMeasure-Theoretic Probability · Mixed review

MTP:010118On {1,2,3,4}, L=(2,1,1,3/4) sends P=(1,2,3,4)/10 to Q=(1/5,1/5,3/10,3/10) with E_Q X=3/5 and TV=1/10

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Measure Theoretic Probability Finite Radon NikodymMeasure Theoretic Probability TV KL Change Measure

Probability measures, random variables, and induced lawsMeasure-Theoretic Probability · Mixed review

MTP:010219Binary Bayes with P(H=1)=1/3 and likelihoods 3/4,1/4 yields posteriors 3/5 and 1/7, E[Var]=6/35, Bayes error 1/4

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Measure Theoretic Probability Binary Bayes PosteriorMeasure Theoretic Probability Total Variance Bayes Error

Conditional expectation and regular conditional structureMeasure-Theoretic Probability · Mixed review

MTP:010222X,Z iid N(0,1) and Y=X+Z give X|Y=y ~ N(y/2,1/2), E[X|Y]=Y/2, MSE=1/2, and P(X>0|Y=2)=Phi(sqrt(2))

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Measure Theoretic Probability Gaussian Conditional LawMeasure Theoretic Probability Regression Mse Decomposition

Conditional expectation and regular conditional structureMeasure-Theoretic Probability · Mixed review

MTP:010320Independent A_n of probability n^{-α} for n≥2: P(i.o.) vanishes iff α>1, and α=2 totals π²/6−1

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Measure Theoretic Probability Borel Cantelli ThresholdMeasure Theoretic Probability Independent Power Series

Independence, product spaces, and zero-one lawsMeasure-Theoretic Probability · Mixed review

MTP:010321Fair independent U,V and W=U xor V are pairwise independent with support 000,011,101,110 each 1/4 and joint entropy 2 bits

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Measure Theoretic Probability Pairwise VS MutualMeasure Theoretic Probability Xor Entropy Support

Independence, product spaces, and zero-one lawsMeasure-Theoretic Probability · Mixed review

MTP:010329Beta(2,1)–Bernoulli mixture has p=2/3, P11=1/2, Cov=1/18, corr=1/4, and predictive (2+s)/(3+s+f)

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Measure Theoretic Probability Beta Bernoulli MixtureMeasure Theoretic Probability Exchangeable Posterior Predictive

Independence, product spaces, and zero-one lawsMeasure-Theoretic Probability · Mixed review

MTP:010423Rademacher harmonic martingale M_n=sum eps_k/k has mean 0, Var(M_3)=49/36, limiting second moment pi^2/6, and diverging absolute series

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Measure Theoretic Probability Rademacher Harmonic MartingaleMeasure Theoretic Probability L2 VS Absolute Series

Modes of convergence and laws of large numbersMeasure-Theoretic Probability · Mixed review

MTP:010524Fair-walk transform T_n=sum S_{k-1}eps_k satisfies S_n^2=n+2T_n, E[T_4^2]=6, and Var(S_4^2)=24

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Measure Theoretic Probability Predictable Transform IdentityMeasure Theoretic Probability Quadratic Variation RW

Martingales, stopping times, and convergenceMeasure-Theoretic Probability · Mixed review

MTP:010525Bernoulli(1/(k+1)) compensator has A_3=13/12, EN_3=13/12, and bracket/variance 95/144, not the mean

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Measure Theoretic Probability Bernoulli CompensatorMeasure Theoretic Probability Inhomogeneous Bracket Variance

Martingales, stopping times, and convergenceMeasure-Theoretic Probability · Mixed review

MTP:010527Fair-walk L_n(ln 2)=2^{S_n}/(5/4)^n is a mean-1 martingale with Q(+)=4/5 and E_P L_n^2=(34/25)^n

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Measure Theoretic Probability Exponential RW MartingaleMeasure Theoretic Probability Likelihood Change Measure

Martingales, stopping times, and convergenceMeasure-Theoretic Probability · Mixed review

MTP:010530Critical binary branching has offspring PGF (1+s^2)/2, Z_n a mean-1 variance-n martingale, q=1, and Z_2 masses 5/8,1/4,1/8

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Measure Theoretic Probability Critical Branching PgfMeasure Theoretic Probability Generation Martingale Extinction

Martingales, stopping times, and convergenceMeasure-Theoretic Probability · Mixed review

MTP:010626Moving atoms delta_{1/n}->delta_0 pass bounded continuous tests, fail the CDF at 0, and have W_p=1/n

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Measure Theoretic Probability Portmanteau Continuity SetsMeasure Theoretic Probability Moving Atom Wasserstein

Weak convergence, characteristic functions, and CLTMeasure-Theoretic Probability · Mixed review

MTP:010628Bin(n,2/n) has PGF limit exp(2(s-1)), mean 2, variance 2(1-2/n), and second factorial moment 4(1-1/n)

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Measure Theoretic Probability Binomial Poisson PgfMeasure Theoretic Probability Rare Event Factorial Moments

Weak convergence, characteristic functions, and CLTMeasure-Theoretic Probability · Mixed review

MTP:010631Exp(1) maxima satisfy P(M_n-log n<=x)=(1-e^{-x}/n)^n->exp(-e^{-x}) with F(0)=e^{-1} and median -ln ln 2

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Measure Theoretic Probability Exponential Maxima GumbelMeasure Theoretic Probability Extreme Value Centering Tightness

Weak convergence, characteristic functions, and CLTMeasure-Theoretic Probability · Mixed review

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MTP:010117Mixed law (1/4)δ0+(3/4)Exp(rate 2) has F(0)=1/4, density (3/2)e^{-2x} on (0,∞), E=3/8, Var=15/64, median (1/2)log(3/2)

MTP:010118On {1,2,3,4}, L=(2,1,1,3/4) sends P=(1,2,3,4)/10 to Q=(1/5,1/5,3/10,3/10) with E_Q X=3/5 and TV=1/10

MTP:010219Binary Bayes with P(H=1)=1/3 and likelihoods 3/4,1/4 yields posteriors 3/5 and 1/7, E[Var]=6/35, Bayes error 1/4

MTP:010222X,Z iid N(0,1) and Y=X+Z give X|Y=y ~ N(y/2,1/2), E[X|Y]=Y/2, MSE=1/2, and P(X>0|Y=2)=Phi(sqrt(2))

MTP:010320Independent A_n of probability n^{-α} for n≥2: P(i.o.) vanishes iff α>1, and α=2 totals π²/6−1

MTP:010321Fair independent U,V and W=U xor V are pairwise independent with support 000,011,101,110 each 1/4 and joint entropy 2 bits

MTP:010329Beta(2,1)–Bernoulli mixture has p=2/3, P11=1/2, Cov=1/18, corr=1/4, and predictive (2+s)/(3+s+f)

MTP:010423Rademacher harmonic martingale M_n=sum eps_k/k has mean 0, Var(M_3)=49/36, limiting second moment pi^2/6, and diverging absolute series

MTP:010524Fair-walk transform T_n=sum S_{k-1}eps_k satisfies S_n^2=n+2T_n, E[T_4^2]=6, and Var(S_4^2)=24

MTP:010525Bernoulli(1/(k+1)) compensator has A_3=13/12, EN_3=13/12, and bracket/variance 95/144, not the mean

MTP:010527Fair-walk L_n(ln 2)=2^{S_n}/(5/4)^n is a mean-1 martingale with Q(+)=4/5 and E_P L_n^2=(34/25)^n

MTP:010530Critical binary branching has offspring PGF (1+s^2)/2, Z_n a mean-1 variance-n martingale, q=1, and Z_2 masses 5/8,1/4,1/8

MTP:010626Moving atoms delta_{1/n}->delta_0 pass bounded continuous tests, fail the CDF at 0, and have W_p=1/n

MTP:010628Bin(n,2/n) has PGF limit exp(2(s-1)), mean 2, variance 2(1-2/n), and second factorial moment 4(1-1/n)

MTP:010631Exp(1) maxima satisfy P(M_n-log n<=x)=(1-e^{-x}/n)^n->exp(-e^{-x}) with F(0)=e^{-1} and median -ln ln 2

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MTP:010117Mixed law (1/4)δ0+(3/4)Exp(rate 2) has F(0)=1/4, density (3/2)e^{-2x} on (0,∞), E=3/8, Var=15/64, median (1/2)log(3/2)

MTP:010118On {1,2,3,4}, L=(2,1,1,3/4) sends P=(1,2,3,4)/10 to Q=(1/5,1/5,3/10,3/10) with E_Q X=3/5 and TV=1/10

MTP:010219Binary Bayes with P(H=1)=1/3 and likelihoods 3/4,1/4 yields posteriors 3/5 and 1/7, E[Var]=6/35, Bayes error 1/4

MTP:010222X,Z iid N(0,1) and Y=X+Z give X|Y=y ~ N(y/2,1/2), E[X|Y]=Y/2, MSE=1/2, and P(X>0|Y=2)=Phi(sqrt(2))

MTP:010320Independent A_n of probability n^{-α} for n≥2: P(i.o.) vanishes iff α>1, and α=2 totals π²/6−1

MTP:010321Fair independent U,V and W=U xor V are pairwise independent with support 000,011,101,110 each 1/4 and joint entropy 2 bits

MTP:010329Beta(2,1)–Bernoulli mixture has p=2/3, P11=1/2, Cov=1/18, corr=1/4, and predictive (2+s)/(3+s+f)

MTP:010423Rademacher harmonic martingale M_n=sum eps_k/k has mean 0, Var(M_3)=49/36, limiting second moment pi^2/6, and diverging absolute series

MTP:010524Fair-walk transform T_n=sum S_{k-1}eps_k satisfies S_n^2=n+2T_n, E[T_4^2]=6, and Var(S_4^2)=24

MTP:010525Bernoulli(1/(k+1)) compensator has A_3=13/12, EN_3=13/12, and bracket/variance 95/144, not the mean

MTP:010527Fair-walk L_n(ln 2)=2^{S_n}/(5/4)^n is a mean-1 martingale with Q(+)=4/5 and E_P L_n^2=(34/25)^n

MTP:010530Critical binary branching has offspring PGF (1+s^2)/2, Z_n a mean-1 variance-n martingale, q=1, and Z_2 masses 5/8,1/4,1/8

MTP:010626Moving atoms delta_{1/n}->delta_0 pass bounded continuous tests, fail the CDF at 0, and have W_p=1/n

MTP:010628Bin(n,2/n) has PGF limit exp(2(s-1)), mean 2, variance 2(1-2/n), and second factorial moment 4(1-1/n)

MTP:010631Exp(1) maxima satisfy P(M_n-log n<=x)=(1-e^{-x}/n)^n->exp(-e^{-x}) with F(0)=e^{-1} and median -ln ln 2

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MTP:010117Mixed law (1/4)δ0+(3/4)Exp(rate 2) has F(0)=1/4, density (3/2)e^{-2x} on (0,∞), E=3/8, Var=15/64, median (1/2)log(3/2)View problem statement

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MTP:010117Mixed law (1/4)δ0+(3/4)Exp(rate 2) has F(0)=1/4, density (3/2)e^{-2x} on (0,∞), E=3/8, Var=15/64, median (1/2)log(3/2)View problem statement

MTP:010117Mixed law (1/4)δ0+(3/4)Exp(rate 2) has F(0)=1/4, density (3/2)e^{-2x} on (0,∞), E=3/8, Var=15/64, median (1/2)log(3/2)

MTP:010117Mixed law (1/4)δ0+(3/4)Exp(rate 2) has F(0)=1/4, density (3/2)e^{-2x} on (0,∞), E=3/8, Var=15/64, median (1/2)log(3/2)