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Cross-validated log Bayes factor for multinomial observations The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #532: "Cross-validated log Bayes factor for multinomial observations" (mult-cvlbf) #NewProof
statproofbook.github.io/P/mult-cvlbf

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Cross-validated log Bayes factor for binomial observations The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #531: "Cross-validated log Bayes factor for binomial observations" (bin-cvlbf) #NewProof
statproofbook.github.io/P/bin-cvlbf

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Partial correlation of random variables which are jointly multivariate normal distributed The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #530: "Partial correlation of random variables which are jointly multivariate normal distributed" (mvn-corrpart) #NewProof
statproofbook.github.io/P/mvn-corrpart

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Conditional correlation of random variables which are jointly multivariate normal distributed The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #529: "Conditional correlation of random variables which are jointly multivariate normal distributed" (mvn-corrcond) #NewProof
statproofbook.github.io/P/mvn-corrcond

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The third standardized moment equals the skewness The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #528: "The third standardized moment equals the skewness" (momstand-3rd) #NewProof
statproofbook.github.io/P/momstand-3rd

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The second standardized moment equals one The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #527: "The second standardized moment equals one" (momstand-2nd) #NewProof
statproofbook.github.io/P/momstand-2nd

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The first standardized moment equals zero The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #526: "The first standardized moment equals zero" (momstand-1st) #NewProof
statproofbook.github.io/P/momstand-1st

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Marginal distributions of the multivariate t-distribution The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #525: "Marginal distributions of the multivariate t-distribution" (mvt-marg) #NewProof
statproofbook.github.io/P/mvt-marg

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Second-order expectations for the matrix-normal distribution The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #524: "Second-order expectations for the matrix-normal distribution" (matn-meanso) #NewProof
statproofbook.github.io/P/matn-meanso

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Expectation of quadratic forms for the matrix-normal distribution The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #523: "Expectation of quadratic forms for the matrix-normal distribution" (matn-meanqf) #NewProof
statproofbook.github.io/P/matn-meanqf

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Cross-covariance matrices of the matrix-normal distribution The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #522: "Cross-covariance matrices of the matrix-normal distribution" (matn-ccm) #NewProof
statproofbook.github.io/P/matn-ccm

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Chi-squared distribution is a special case of gamma distribution The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #521: "Chi-squared distribution is a special case of gamma distribution" (chi2-gam2) #NewProof
statproofbook.github.io/P/chi2-gam2

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Mode of the beta distribution The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #520: "Mode of the beta distribution" (beta-mode) #NewProof
statproofbook.github.io/P/beta-mode

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Median of the beta distribution The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #519: "Median of the beta distribution" (beta-med) #NewProof
statproofbook.github.io/P/beta-med

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Median of the gamma distribution The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #518: "Median of the gamma distribution" (gam-med) #NewProof
statproofbook.github.io/P/gam-med

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The variational free energy is a lower bound on the log model evidence The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #517: "The variational free energy is a lower bound on the log model evidence" (fren-lme) #NewProof
statproofbook.github.io/P/fren-lme

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Decompositions of the variational free energy The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #516: "Decompositions of the variational free energy" (fren-dec) #NewProof
statproofbook.github.io/P/fren-dec

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Non-negativity of the Kullback-Leibler divergence The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #515: "Non-negativity of the Kullback-Leibler divergence" (kl-nonneg3) #NewProof
statproofbook.github.io/P/kl-nonneg3

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Jensen’s inequality The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #514: "Jensen's inequality" (jens-ineq) #NewProof
statproofbook.github.io/P/jens-ineq

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Expectation of a bilinear form The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #513: "Expectation of a bilinear form" (mean-blf) #NewProof
statproofbook.github.io/P/mean-blf

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Partition of a cross-covariance matrix into expected values The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #512: "Partition of a cross-covariance matrix into expected values" (covmatcross-mean) #NewProof
statproofbook.github.io/P/covmatcros...

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Prior predictive distribution is a marginal distribution of the joint likelihood The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #510: "Prior predictive distribution is a marginal distribution of the joint likelihood" (priorpred-jl) #NewProof
statproofbook.github.io/P/priorpred-jl

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Variance of the coefficient of determination under the null hypothesis The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #509: "Variance of the coefficient of determination under the null hypothesis" (rsq-var) #NewProof
statproofbook.github.io/P/rsq-var

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Mean, mode and median of the coefficient of determination under the null hypothesis The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #508: "Mean, mode and median of the coefficient of determination under the null hypothesis" (rsq-mmm) #NewProof
statproofbook.github.io/P/rsq-mmm

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Distribution of the coefficient of determination under the null hypothesis The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #507: "Distribution of the coefficient of determination under the null hypothesis" (rsq-dist) #NewProof
statproofbook.github.io/P/rsq-dist

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Redundancy of parameters describing the matrix-normal distribution The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #506: "Redundancy of parameters describing the matrix-normal distribution" (matn-red2) #NewProof
statproofbook.github.io/P/matn-red2

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Redundancy of parameters describing the matrix-normal distribution The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #505: "Redundancy of parameters describing the matrix-normal distribution" (matn-red) #NewProof
statproofbook.github.io/P/matn-red

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Maximum likelihood estimation for multivariate normally distributed data The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #504: "Maximum likelihood estimation for multivariate normally distributed data" (mvn-mle) #NewProof
statproofbook.github.io/P/mvn-mle

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Maximum likelihood estimation for bivariate normally distributed data The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #503: "Maximum likelihood estimation for bivariate normally distributed data" (bvn-mle) #NewProof
statproofbook.github.io/P/bvn-mle

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Construction of a bivariate normal distribution from univariate standard normal random variables The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences

Proof #502: "Construction of a bivariate normal distribution from univariate standard normal random variables" (bvn-snorm) #NewProof
statproofbook.github.io/P/bvn-snorm

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