Merge pull request #96 from StatProofBook/master

update to master

Author: JoramSoch

I/PbA.md

@@ -8,7 +8,7 @@ title: "Proof by Author"
 
 - [Covariance matrix of the multinomial distribution](/P/mult-cov)
 
-### JoramSoch (336 proofs)
+### JoramSoch (339 proofs)
 
 - [Accuracy and complexity for the univariate Gaussian](/P/ug-anc)
 - [Accuracy and complexity for the univariate Gaussian with known variance](/P/ugkv-anc)
@@ -48,8 +48,8 @@ title: "Proof by Author"
 - [Covariance matrices of the matrix-normal distribution](/P/matn-cov)
 - [Covariance matrix of the categorical distribution](/P/cat-cov)
 - [Covariance matrix of the multivariate normal distribution](/P/mvn-cov)
+- [Covariance matrix of the sum of two random vectors](/P/covmat-sum)
 - [Covariance of independent random variables](/P/cov-ind)
-- [Covariance of the sum of two random vectors](/P/covmat-sum)
 - [Cross-validated log Bayes factor for the univariate Gaussian with known variance](/P/ugkv-cvlbf)
 - [Cross-validated log model evidence for the univariate Gaussian with known variance](/P/ugkv-cvlme)
 - [Cumulative distribution function in terms of probability density function of a continuous random variable](/P/cdf-pdf)
@@ -146,6 +146,7 @@ title: "Proof by Author"
 - [Log sum inequality](/P/logsum-ineq)
 - [Log-odds and probability in logistic regression](/P/logreg-lonp)
 - [Logarithmic expectation of the gamma distribution](/P/gam-logmean)
+- [Marginal distribution of a conditional binomial distribution](/P/bin-margcond)
 - [Marginal distributions for the matrix-normal distribution](/P/matn-marg)
 - [Marginal distributions of the multivariate normal distribution](/P/mvn-marg)
 - [Marginal distributions of the normal-gamma distribution](/P/ng-marg)
@@ -154,8 +155,8 @@ title: "Proof by Author"
 - [Maximum likelihood estimation for Dirichlet-distributed data](/P/dir-mle)
 - [Maximum likelihood estimation for multiple linear regression](/P/mlr-mle)
 - [Maximum likelihood estimation for Poisson-distributed data](/P/poiss-mle)
-- [Maximum likelihood estimation for simple linear regression](/P/slr-mle2)
 - [Maximum likelihood estimation for simple linear regression](/P/slr-mle)
+- [Maximum likelihood estimation for simple linear regression](/P/slr-mle2)
 - [Maximum likelihood estimation for the general linear model](/P/glm-mle)
 - [Maximum likelihood estimation for the Poisson distribution with exposure values](/P/poissexp-mle)
 - [Maximum likelihood estimation for the univariate Gaussian](/P/ug-mle)
@@ -179,6 +180,7 @@ title: "Proof by Author"
 - [Median of the continuous uniform distribution](/P/cuni-med)
 - [Median of the exponential distribution](/P/exp-med)
 - [Median of the normal distribution](/P/norm-med)
+- [Method of moments for beta-binomial data](/P/betabin-mome)
 - [Method of moments for beta-distributed data](/P/beta-mome)
 - [Mode of the continuous uniform distribution](/P/cuni-med)
 - [Mode of the normal distribution](/P/norm-mode)
@@ -195,8 +197,8 @@ title: "Proof by Author"
 - [Non-invariance of the differential entropy under change of variables](/P/dent-noninv)
 - [(Non-)Multiplicativity of the expected value](/P/mean-mult)
 - [Non-negativity of the expected value](/P/mean-nonneg)
-- [Non-negativity of the Kullback-Leibler divergence](/P/kl-nonneg2)
 - [Non-negativity of the Kullback-Leibler divergence](/P/kl-nonneg)
+- [Non-negativity of the Kullback-Leibler divergence](/P/kl-nonneg2)
 - [Non-negativity of the Shannon entropy](/P/ent-nonneg)
 - [Non-negativity of the variance](/P/var-nonneg)
 - [Non-symmetry of the Kullback-Leibler divergence](/P/kl-nonsymm)
@@ -207,8 +209,8 @@ title: "Proof by Author"
 - [One-sample z-test for independent observations](/P/ugkv-ztest1)
 - [Ordinary least squares for multiple linear regression](/P/mlr-ols)
 - [Ordinary least squares for multiple linear regression](/P/mlr-ols2)
-- [Ordinary least squares for simple linear regression](/P/slr-ols2)
 - [Ordinary least squares for simple linear regression](/P/slr-ols)
+- [Ordinary least squares for simple linear regression](/P/slr-ols2)
 - [Ordinary least squares for the general linear model](/P/glm-ols)
 - [Paired t-test for dependent observations](/P/ug-ttestp)
 - [Paired z-test for dependent observations](/P/ugkv-ztestp)
@@ -293,6 +295,7 @@ title: "Proof by Author"
 - [Relation of mutual information to joint and conditional entropy](/P/dmi-jce)
 - [Relation of mutual information to marginal and conditional entropy](/P/dmi-mce)
 - [Relation of mutual information to marginal and joint entropy](/P/dmi-mje)
+- [Relationship between chi-squared distribution and beta distribution](/P/beta-chi2)
 - [Relationship between coefficient of determination and correlation coefficient in simple linear regression](/P/slr-rsq)
 - [Relationship between correlation coefficient and slope estimate in simple linear regression](/P/slr-corr)
 - [Relationship between covariance and correlation](/P/cov-corr)
@@ -341,10 +344,10 @@ title: "Proof by Author"
 - [Variance of the normal distribution](/P/norm-var)
 - [Variance of the Poisson distribution](/P/poiss-var)
 - [Variance of the sum of two random variables](/P/var-sum)
-- [Weighted least squares for multiple linear regression](/P/mlr-wls2)
 - [Weighted least squares for multiple linear regression](/P/mlr-wls)
-- [Weighted least squares for simple linear regression](/P/slr-wls2)
+- [Weighted least squares for multiple linear regression](/P/mlr-wls2)
 - [Weighted least squares for simple linear regression](/P/slr-wls)
+- [Weighted least squares for simple linear regression](/P/slr-wls2)
 - [Weighted least squares for the general linear model](/P/glm-wls)
 
 ### kantundpeterpan (1 proof)

I/PbN.md

@@ -354,10 +354,13 @@ title: "Proof by Number"
 | P346 | ng-samp | [Sampling from the normal-gamma distribution](/P/ng-samp) | JoramSoch | 2022-09-22 |
 | P347 | covmat-inv | [Invariance of the covariance matrix under addition of constant vector](/P/covmat-inv) | JoramSoch | 2022-09-22 |
 | P348 | covmat-scal | [Scaling of the covariance matrix upon multiplication with constant matrix](/P/covmat-scal) | JoramSoch | 2022-09-22 |
-| P349 | covmat-sum | [Covariance of the sum of two random vectors](/P/covmat-sum) | JoramSoch | 2022-09-26 |
+| P349 | covmat-sum | [Covariance matrix of the sum of two random vectors](/P/covmat-sum) | JoramSoch | 2022-09-26 |
 | P350 | covmat-symm | [Symmetry of the covariance matrix](/P/covmat-symm) | JoramSoch | 2022-09-26 |
 | P351 | covmat-psd | [Positive semi-definiteness of the covariance matrix](/P/covmat-psd) | JoramSoch | 2022-09-26 |
 | P352 | cov-var | [Self-covariance equals variance](/P/cov-var) | JoramSoch | 2022-09-26 |
 | P353 | cov-symm | [Symmetry of the covariance](/P/cov-symm) | JoramSoch | 2022-09-26 |
 | P354 | lognorm-mean | [Mean of the log-normal distribution](/P/lognorm-mean) | majapavlo | 2022-10-02 |
 | P355 | lognorm-var | [Variance of the log-normal distribution](/P/lognorm-var) | majapavlo | 2022-10-02 |
+| P356 | beta-chi2 | [Relationship between chi-squared distribution and beta distribution](/P/beta-chi2) | JoramSoch | 2022-10-07 |
+| P357 | betabin-mome | [Method of moments for beta-binomial data](/P/betabin-mome) | JoramSoch | 2022-10-07 |
+| P358 | bin-margcond | [Marginal distribution of a conditional binomial distribution](/P/bin-margcond) | JoramSoch | 2022-10-07 |

I/PbT.md

@@ -52,8 +52,8 @@ title: "Proof by Topic"
 - [Covariance matrix of the categorical distribution](/P/cat-cov)
 - [Covariance matrix of the multinomial distribution](/P/mult-cov)
 - [Covariance matrix of the multivariate normal distribution](/P/mvn-cov)
+- [Covariance matrix of the sum of two random vectors](/P/covmat-sum)
 - [Covariance of independent random variables](/P/cov-ind)
-- [Covariance of the sum of two random vectors](/P/covmat-sum)
 - [Cross-validated log Bayes factor for the univariate Gaussian with known variance](/P/ugkv-cvlbf)
 - [Cross-validated log model evidence for the univariate Gaussian with known variance](/P/ugkv-cvlme)
 - [Cumulative distribution function in terms of probability density function of a continuous random variable](/P/cdf-pdf)
@@ -179,6 +179,7 @@ title: "Proof by Topic"
 
 ### M
 
+- [Marginal distribution of a conditional binomial distribution](/P/bin-margcond)
 - [Marginal distributions for the matrix-normal distribution](/P/matn-marg)
 - [Marginal distributions of the multivariate normal distribution](/P/mvn-marg)
 - [Marginal distributions of the normal-gamma distribution](/P/ng-marg)
@@ -187,8 +188,8 @@ title: "Proof by Topic"
 - [Maximum likelihood estimation for Dirichlet-distributed data](/P/dir-mle)
 - [Maximum likelihood estimation for multiple linear regression](/P/mlr-mle)
 - [Maximum likelihood estimation for Poisson-distributed data](/P/poiss-mle)
-- [Maximum likelihood estimation for simple linear regression](/P/slr-mle2)
 - [Maximum likelihood estimation for simple linear regression](/P/slr-mle)
+- [Maximum likelihood estimation for simple linear regression](/P/slr-mle2)
 - [Maximum likelihood estimation for the general linear model](/P/glm-mle)
 - [Maximum likelihood estimation for the Poisson distribution with exposure values](/P/poissexp-mle)
 - [Maximum likelihood estimation for the univariate Gaussian](/P/ug-mle)
@@ -215,6 +216,7 @@ title: "Proof by Topic"
 - [Median of the exponential distribution](/P/exp-med)
 - [Median of the log-normal distribution](/P/lognorm-med)
 - [Median of the normal distribution](/P/norm-med)
+- [Method of moments for beta-binomial data](/P/betabin-mome)
 - [Method of moments for beta-distributed data](/P/beta-mome)
 - [Mode of the continuous uniform distribution](/P/cuni-med)
 - [Mode of the exponential distribution](/P/exp-mode)
@@ -238,8 +240,8 @@ title: "Proof by Topic"
 - [Non-invariance of the differential entropy under change of variables](/P/dent-noninv)
 - [(Non-)Multiplicativity of the expected value](/P/mean-mult)
 - [Non-negativity of the expected value](/P/mean-nonneg)
-- [Non-negativity of the Kullback-Leibler divergence](/P/kl-nonneg2)
 - [Non-negativity of the Kullback-Leibler divergence](/P/kl-nonneg)
+- [Non-negativity of the Kullback-Leibler divergence](/P/kl-nonneg2)
 - [Non-negativity of the Shannon entropy](/P/ent-nonneg)
 - [Non-negativity of the variance](/P/var-nonneg)
 - [Non-symmetry of the Kullback-Leibler divergence](/P/kl-nonsymm)
@@ -253,8 +255,8 @@ title: "Proof by Topic"
 - [One-sample z-test for independent observations](/P/ugkv-ztest1)
 - [Ordinary least squares for multiple linear regression](/P/mlr-ols)
 - [Ordinary least squares for multiple linear regression](/P/mlr-ols2)
-- [Ordinary least squares for simple linear regression](/P/slr-ols2)
 - [Ordinary least squares for simple linear regression](/P/slr-ols)
+- [Ordinary least squares for simple linear regression](/P/slr-ols2)
 - [Ordinary least squares for the general linear model](/P/glm-ols)
 
 ### P
@@ -352,6 +354,7 @@ title: "Proof by Topic"
 - [Relation of mutual information to joint and conditional entropy](/P/dmi-jce)
 - [Relation of mutual information to marginal and conditional entropy](/P/dmi-mce)
 - [Relation of mutual information to marginal and joint entropy](/P/dmi-mje)
+- [Relationship between chi-squared distribution and beta distribution](/P/beta-chi2)
 - [Relationship between coefficient of determination and correlation coefficient in simple linear regression](/P/slr-rsq)
 - [Relationship between correlation coefficient and slope estimate in simple linear regression](/P/slr-corr)
 - [Relationship between covariance and correlation](/P/cov-corr)
@@ -416,8 +419,8 @@ title: "Proof by Topic"
 
 ### W
 
-- [Weighted least squares for multiple linear regression](/P/mlr-wls2)
 - [Weighted least squares for multiple linear regression](/P/mlr-wls)
-- [Weighted least squares for simple linear regression](/P/slr-wls2)
+- [Weighted least squares for multiple linear regression](/P/mlr-wls2)
 - [Weighted least squares for simple linear regression](/P/slr-wls)
+- [Weighted least squares for simple linear regression](/P/slr-wls2)
 - [Weighted least squares for the general linear model](/P/glm-wls)

I/PwS.md

@@ -55,8 +55,8 @@ title: "Proofs without Source"
 - [Maximum likelihood estimation can result in biased estimates](/P/mle-bias)
 - [Maximum likelihood estimation for multiple linear regression](/P/mlr-mle)
 - [Maximum likelihood estimation for Poisson-distributed data](/P/poiss-mle)
-- [Maximum likelihood estimation for simple linear regression](/P/slr-mle2)
 - [Maximum likelihood estimation for simple linear regression](/P/slr-mle)
+- [Maximum likelihood estimation for simple linear regression](/P/slr-mle2)
 - [Maximum likelihood estimation for the general linear model](/P/glm-mle)
 - [Maximum likelihood estimation for the Poisson distribution with exposure values](/P/poissexp-mle)
 - [Mean of the categorical distribution](/P/cat-mean)
@@ -117,6 +117,6 @@ title: "Proofs without Source"
 - [Transposition of a matrix-normal random variable](/P/matn-trans)
 - [Variance of the Wald distribution](/P/wald-var)
 - [Weighted least squares for multiple linear regression](/P/mlr-wls2)
-- [Weighted least squares for simple linear regression](/P/slr-wls2)
 - [Weighted least squares for simple linear regression](/P/slr-wls)
+- [Weighted least squares for simple linear regression](/P/slr-wls2)
 - [Weighted least squares for the general linear model](/P/glm-wls)

I/ToC.md

@@ -424,7 +424,7 @@ title: "Table of Contents"
    &emsp;&ensp; 3.6.2. **[Probability density function](/P/lognorm-pdf)** <br>
    &emsp;&ensp; 3.6.3. **[Cumulative distribution function](/P/lognorm-cdf)** <br>
    &emsp;&ensp; 3.6.4. **[Quantile Function](/P/lognorm-qf)** <br>
-   &emsp;&ensp; 3.6.5. **[Mean](/P/lognorm-mean)** <br>   
+   &emsp;&ensp; 3.6.5. **[Mean](/P/lognorm-mean)** <br>
    &emsp;&ensp; 3.6.6. **[Median](/P/lognorm-med)** <br>
    &emsp;&ensp; 3.6.7. **[Mode](/P/lognorm-mode)** <br>
    &emsp;&ensp; 3.6.8. **[Variance](/P/lognorm-var)** <br>

P/betabin-mome.md

@@ -36,7 +36,7 @@ username: "JoramSoch"
 **Theorem:** Let $y = \left\lbrace y_1, \ldots, y_N \right\rbrace$ be a set of observed counts independent and identically distributed according to a [beta-binomial distribution](/D/betabin) with number of trials $n$ as well as parameters $\alpha$ and $\beta$:
 
 $$ \label{eq:binbeta}
-y_i \sim \mathrm{BinBet}(n, \alpha, \beta), \quad i = 1, \ldots, N \; .
+y_i \sim \mathrm{BetBin}(n, \alpha, \beta), \quad i = 1, \ldots, N \; .
 $$
 
 Then, the [method-of-moments estimates](/D/mome) for the parameters $\alpha$ and $\beta$ are given by

P/bin-margcond.md

@@ -91,7 +91,7 @@ $$ \label{eq:bin-th}
 (x+y)^n = \sum_{k=0}^{n} {n \choose k} \, x^{n-k} \, y^k \; ,
 $$
 
-the sum is equal to
+the sum in equation \eqref{eq:Y-dist-s6} is equal to
 
 $$ \label{eq:Y-dist-sum}
 \sum_{i=0}^{n-m} {n-m \choose i} \, \left( p - pq \right)^{i} \, (1-p)^{n-m-i} = \left( (p-pq)+(1-p) \right)^{n-m} \; .

P/covmat-sum.md

@@ -7,7 +7,7 @@ affiliation: "BCCN Berlin"
 e_mail: "joram.soch@bccn-berlin.de"
 date: 2022-09-26 10:37:00
 
-title: "Covariance of the sum of two random vectors"
+title: "Covariance matrix of the sum of two random vectors"
 chapter: "General Theorems"
 section: "Probability theory"
 topic: "Covariance"

P/f-pdf.md

@@ -94,7 +94,7 @@ $$ \label{eq:f-XY}
 f_{X,Y}(x,y) = f_X(x) \cdot f_Y(y) \; .
 $$
 
-With the [probability density function of an invertible function](/P/pdf-invfct), the [joint density](/D/dist-joint) of $T$ and $W$ can be derived as:
+With the [probability density function of an invertible function](/P/pdf-invfct), the [joint density](/D/dist-joint) of $F$ and $W$ can be derived as:
 
 $$ \label{eq:f-FW-s1}
 f_{F,W}(f,w) = f_{X,Y}(x,y) \cdot \lvert J \rvert \; .

P/lognorm-cdf.md

@@ -66,7 +66,7 @@ From this point forward, the proof is similar to the derivation of the [cumulati
 
 $$ \label{eq:lognorm-cdf-s2}
 \begin{split}
-F_X(x) &= \frac{1}{\sigma \sqrt{2 \pi}} \int_{(-\infty-\mu)/(\sqrt{2} \sigma)}^{(\ln x-\mu)/(\sqrt{2} \sigma)} \frac{1}{\exp( \sqrt{2} \sigma t + \mu)} \cdot \exp \left(-t^2 \right) \, \mathrm{d} \left( \exp \left( \sqrt{2} \sigma t + \mu \right) \right) \\
+F_X(x) &= \frac{1}{\sigma \sqrt{2 \pi}} \int_{(-\infty-\mu)/(\sqrt{2} \sigma)}^{(\ln x-\mu)/(\sqrt{2} \sigma)} \frac{1}{\exp( \sqrt{2} \sigma t + \mu)} \cdot \exp \left(-t^2 \right) \, \mathrm{d} \left[ \exp \left( \sqrt{2} \sigma t + \mu \right) \right] \\
 &=\frac{\sqrt{2} \sigma}{\sigma \sqrt{2 \pi}} \int_{-\infty}^{\frac{\ln x-\mu}{\sqrt{2} \sigma}} \frac{1}{\exp( \sqrt{2} \sigma t + \mu)} \cdot
 \exp(-t^2) \cdot \exp \left( \sqrt{2} \sigma t + \mu \right) \, \mathrm{d}t  \\
 &= \frac{1}{\sqrt{\pi}} \int_{-\infty}^{\frac{\ln x-\mu}{\sqrt{2} \sigma}} \exp(-t^2) \, \mathrm{d}t \\