The Book of Statistical Proofs
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  • Accuracy and complexity for the univariate Gaussian
  • Accuracy and complexity for the univariate Gaussian with known variance
  • Bayesian information criterion for multiple linear regression
  • Bernoulli distribution is a special case of categorical distribution
  • Beta distribution is a special case of Dirichlet distribution
  • Binomial distribution is a special case of multinomial distribution
  • Chi-squared distribution is a special case of gamma distribution
  • Combined posterior distribution for Bayesian linear regression when analyzing conditionally independent data sets
  • Combined posterior distributions in terms of individual posterior distributions obtained from conditionally independent data
  • Conditional distributions of the normal-gamma distribution
  • Construction of unbiased estimator for variance in multiple linear regression
  • Covariance and variance of the normal-gamma distribution
  • Covariance matrix of the categorical distribution
  • Cross-validated log Bayes factor for the univariate Gaussian
  • Cross-validated log Bayes factor for the univariate Gaussian with known variance
  • Cross-validated log model evidence for binomial observations
  • Cross-validated log model evidence for multinomial observations
  • Cross-validated log model evidence for the univariate Gaussian
  • Cross-validated log model evidence for the univariate Gaussian with known variance
  • Cumulative distribution function in terms of probability density function of a continuous random variable
  • Cumulative distribution function in terms of probability mass function of a discrete random variable
  • Cumulative distribution function of the beta-binomial distribution
  • Cumulative distribution function of the continuous uniform distribution
  • Cumulative distribution function of the discrete uniform distribution
  • Cumulative distribution function of the exponential distribution
  • Cumulative distribution function of the multinomial distribution
  • Derivation of Bayesian model averaging
  • Derivation of the family evidence
  • Derivation of the log Bayes factor
  • Derivation of the log family evidence
  • Derivation of the log model evidence
  • Derivation of the model evidence
  • Derivation of the posterior model probability
  • Deviance for multiple linear regression
  • Deviance information criterion for multiple linear regression
  • Differential entropy for the matrix-normal distribution
  • Differential entropy of the continuous uniform distribution
  • Differential entropy of the normal-gamma distribution
  • Effects of mean-centering on parameter estimates for simple linear regression
  • Entropy of the binomial distribution
  • Entropy of the categorical distribution
  • Entropy of the discrete uniform distribution
  • Entropy of the multinomial distribution
  • Entropy of the Poisson distribution
  • Expectation of the cross-validated log Bayes factor for the univariate Gaussian with known variance
  • Expectation of the log Bayes factor for the univariate Gaussian with known variance
  • Exponential distribution is a special case of gamma distribution
  • Expression of R² in terms of residual variances
  • Expression of the noise precision posterior for Bayesian linear regression using prediction and parameter errors
  • F-statistic in terms of ordinary least squares estimates in one-way analysis of variance
  • F-statistics in terms of ordinary least squares estimates in two-way analysis of variance
  • First raw moment is mean
  • Gamma distribution is a special case of Wishart distribution
  • Joint likelihood is the product of likelihood function and prior density
  • Kullback-Leibler divergence for the Bernoulli distribution
  • Kullback-Leibler divergence for the continuous uniform distribution
  • Kullback-Leibler divergence for the discrete uniform distribution
  • Kullback-Leibler divergence for the matrix-normal distribution
  • Kullback-Leibler divergence for the normal distribution
  • Linear combination of independent normal random variables
  • Linear transformation theorem for the matrix-normal distribution
  • Log Bayes factor for Bayesian linear regression
  • Log Bayes factor for binomial observations
  • Log Bayes factor for multinomial observations
  • Log Bayes factor for the univariate Gaussian with known variance
  • Log model evidence for multinomial observations
  • Log model evidence for multivariate Bayesian linear regression
  • Log model evidence for Poisson-distributed data
  • Log model evidence for the Poisson distribution with exposure values
  • Log model evidence for the univariate Gaussian with known variance
  • Log model evidence in terms of prior and posterior distribution
  • Log-likelihood ratio for the general linear model
  • Marginal distributions for the matrix-normal distribution
  • Marginal distributions of the multinomial distribution
  • Marginal distributions of the multivariate normal distribution
  • Marginal distributions of the normal-gamma distribution
  • Marginal likelihood is a definite integral of the joint likelihood
  • Maximum likelihood estimation can result in biased estimates
  • Maximum likelihood estimation for multinomial observations
  • Maximum likelihood estimation for multiple linear regression
  • Maximum likelihood estimation for Poisson-distributed data
  • Maximum likelihood estimation for simple linear regression
  • Maximum likelihood estimation for simple linear regression
  • Maximum likelihood estimation for the Poisson distribution with exposure values
  • Maximum log-likelihood for binomial observations
  • Maximum log-likelihood for multinomial observations
  • Maximum log-likelihood for the general linear model
  • Maximum-a-posteriori estimation for Bayesian linear regression
  • Maximum-a-posteriori estimation for binomial observations
  • Maximum-a-posteriori estimation for multinomial observations
  • Mean of the categorical distribution
  • Mean of the continuous uniform distribution
  • Mean of the ex-Gaussian distribution
  • Mean of the multinomial distribution
  • Mean of the normal-gamma distribution
  • Mean of the normal-Wishart distribution
  • Mean of the Wald distribution
  • Median of the continuous uniform distribution
  • Median of the exponential distribution
  • Median of the log-normal distribution
  • Median of the normal distribution
  • Method of moments for ex-Gaussian-distributed data
  • Method of moments for Wald-distributed data
  • Mode of the continuous uniform distribution
  • Mode of the exponential distribution
  • Mode of the gamma distribution
  • Mode of the normal distribution
  • Moment-generating function of the ex-Gaussian distribution
  • Moment-generating function of the exponential distribution
  • Moment-generating function of the gamma distribution
  • Moment-generating function of the multivariate normal distribution
  • Mutual information of dependent and independent variables in the general linear model
  • Necessary and sufficient condition for independence of multivariate normal random variables
  • Normal-gamma distribution is a special case of normal-Wishart distribution
  • Ordinary least squares for multiple linear regression with two regressors
  • Ordinary least squares for one-way analysis of variance
  • Ordinary least squares for simple linear regression
  • Ordinary least squares for the general linear model
  • Parameter estimates for simple linear regression are uncorrelated after mean-centering
  • Posterior density is proportional to joint likelihood
  • Posterior predictive distribution is a marginal distribution of the joint likelihood
  • Posterior probability of the alternative model for binomial observations
  • Posterior probability of the alternative model for multinomial observations
  • Probability density function of the beta distribution
  • Probability density function of the bivariate normal distribution
  • Probability density function of the continuous uniform distribution
  • Probability density function of the Dirichlet distribution
  • Probability density function of the ex-Gaussian distribution
  • Probability density function of the exponential distribution
  • Probability density function of the gamma distribution
  • Probability density function of the matrix-normal distribution
  • Probability density function of the multivariate normal distribution
  • Probability density function of the multivariate t-distribution
  • Probability density function of the normal distribution
  • Probability density function of the Wald distribution
  • Probability mass function of the Bernoulli distribution
  • Probability mass function of the binomial distribution
  • Probability mass function of the categorical distribution
  • Probability mass function of the discrete uniform distribution
  • Probability mass function of the multinomial distribution
  • Probability mass function of the Poisson distribution
  • Quantile function of the continuous uniform distribution
  • Quantile function of the discrete uniform distribution
  • Quantile function of the exponential distribution
  • Range of the variance of the binomial distribution
  • Relationship between covariance and correlation
  • Relationship between F-statistic and maximum log-likelihood
  • Relationship between gamma distribution and standard gamma distribution
  • Relationship between gamma distribution and standard gamma distribution
  • Relationship between non-standardized t-distribution and t-distribution
  • Relationship between normal distribution and standard normal distribution
  • Relationship between normal distribution and standard normal distribution
  • Relationship between normal distribution and standard normal distribution
  • Relationship between precision matrix and correlation matrix
  • Relationship between R² and maximum log-likelihood
  • Relationship between second raw moment, variance and mean
  • Relationship between signal-to-noise ratio and maximum log-likelihood
  • Relationship between signal-to-noise ratio and R²
  • Scaling of a random variable following the gamma distribution
  • Simple linear regression is a special case of multiple linear regression
  • Skewness of the ex-Gaussian distribution
  • Skewness of the exponential distribution
  • Skewness of the Wald distribution
  • Statistical test for comparing simple linear regression models with and without slope parameter
  • Statistical test for intercept parameter in simple linear regression model
  • Statistical test for slope parameter in simple linear regression model
  • Sums of squares for simple linear regression
  • Transformation matrices for simple linear regression
  • Transitivity of Bayes Factors
  • Transposition of a matrix-normal random variable
  • Variance of the continuous uniform distribution
  • Variance of the ex-Gaussian distribution
  • Variance of the Wald distribution
  • Weighted least squares for multiple linear regression
  • Weighted least squares for simple linear regression
  • Weighted least squares for simple linear regression
  • Weighted least squares for the general linear model
  • The Book of Statistical Proofs
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  • StatProofBook
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The Book of Statistical Proofs – a centralized, open and collaboratively edited archive of statistical theorems for the computational sciences; available under CC-BY-SA 4.0.