Merge pull request #93 from StatProofBook/master

JoramSoch · web-flow · commit 6aa01e5b9444 · 2022-09-22T11:51:14.000+02:00
update to master
diff --git a/D/mult.md b/D/mult.md
@@ -33,4 +33,4 @@ $$ \label{eq:mult}
 X \sim \mathrm{Mult}(n, \left[p_1, \ldots, p_k \right]) \; ,
 $$
 
-if $X$ are the numbers of observations belonging to $k$ distinct categories in $n$ [independent](/D/ind) trials, where each trial has [$k$ possible outcomes](/D/cat) and the category probabilities are identical across trials.
+if $X$ are the numbers of observations belonging to $k$ distinct categories in $n$ [independent](/D/ind) trials, where each trial has $k$ [possible outcomes](/D/cat) and the category probabilities are identical across trials.
diff --git a/I/PbA.md b/I/PbA.md
@@ -8,7 +8,7 @@ title: "Proof by Author"
 
 - [Covariance matrix of the multinomial distribution](/P/mult-cov)
 
-### JoramSoch (321 proofs)
+### JoramSoch (326 proofs)
 
 - [Accuracy and complexity for the univariate Gaussian](/P/ug-anc)
 - [Accuracy and complexity for the univariate Gaussian with known variance](/P/ugkv-anc)
@@ -44,7 +44,9 @@ title: "Proof by Author"
 - [Corrected Akaike information criterion in terms of maximum log-likelihood](/P/aicc-mll)
 - [Correlation always falls between -1 and +1](/P/corr-range)
 - [Correlation coefficient in terms of standard scores](/P/corr-z)
+- [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 of independent random variables](/P/cov-ind)
 - [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)
@@ -140,6 +142,7 @@ 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 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)
 - [Marginal likelihood is a definite integral of joint likelihood](/P/ml-jl)
@@ -162,7 +165,9 @@ title: "Proof by Author"
 - [Mean of the continuous uniform distribution](/P/cuni-mean)
 - [Mean of the exponential distribution](/P/exp-mean)
 - [Mean of the gamma distribution](/P/gam-mean)
+- [Mean of the matrix-normal distribution](/P/matn-mean)
 - [Mean of the multinomial distribution](/P/mult-mean)
+- [Mean of the multivariate normal distribution](/P/mvn-mean)
 - [Mean of the normal distribution](/P/norm-mean)
 - [Mean of the normal-gamma distribution](/P/ng-mean)
 - [Mean of the normal-Wishart distribution](/P/nw-mean)
@@ -279,7 +284,7 @@ title: "Proof by Author"
 - [Relation of continuous mutual information to joint and conditional differential entropy](/P/cmi-jcde)
 - [Relation of continuous mutual information to marginal and conditional differential entropy](/P/cmi-mcde)
 - [Relation of continuous mutual information to marginal and joint differential entropy](/P/cmi-mjde)
-- [Relation of Kullback-Leibler divergence to entropy](/P/kl-ent)
+- [Relation of discrete Kullback-Leibler divergence to Shannon entropy](/P/kl-ent)
 - [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)
diff --git a/I/PbN.md b/I/PbN.md
@@ -119,7 +119,7 @@ title: "Proof by Number"
 | P110 | gam-logmean | [Logarithmic expectation of the gamma distribution](/P/gam-logmean) | JoramSoch | 2020-05-25 |
 | P111 | norm-snorm | [Relationship between normal distribution and standard normal distribution](/P/norm-snorm) | JoramSoch | 2020-05-26 |
 | P112 | gam-sgam | [Relationship between gamma distribution and standard gamma distribution](/P/gam-sgam) | JoramSoch | 2020-05-26 |
-| P113 | kl-ent | [Relation of Kullback-Leibler divergence to entropy](/P/kl-ent) | JoramSoch | 2020-05-27 |
+| P113 | kl-ent | [Relation of discrete Kullback-Leibler divergence to Shannon entropy](/P/kl-ent) | JoramSoch | 2020-05-27 |
 | P114 | kl-dent | [Relation of continuous Kullback-Leibler divergence to differential entropy](/P/kl-dent) | JoramSoch | 2020-05-27 |
 | P115 | kl-inv | [Invariance of the Kullback-Leibler divergence under parameter transformation](/P/kl-inv) | JoramSoch | 2020-05-28 |
 | P116 | kl-add | [Additivity of the Kullback-Leibler divergence for independent distributions](/P/kl-add) | JoramSoch | 2020-05-31 |
@@ -344,3 +344,8 @@ title: "Proof by Number"
 | P336 | cat-ent | [Entropy of the categorical distribution](/P/cat-ent) | JoramSoch | 2022-09-09 |
 | P337 | mult-ent | [Entropy of the multinomial distribution](/P/mult-ent) | JoramSoch | 2022-09-09 |
 | P338 | cat-cov | [Covariance matrix of the categorical distribution](/P/cat-cov) | JoramSoch | 2022-09-09 |
+| P339 | mvn-mean | [Mean of the multivariate normal distribution](/P/mvn-mean) | JoramSoch | 2022-09-15 |
+| P340 | mvn-cov | [Covariance matrix of the multivariate normal distribution](/P/mvn-cov) | JoramSoch | 2022-09-15 |
+| P341 | matn-mean | [Mean of the matrix-normal distribution](/P/matn-mean) | JoramSoch | 2022-09-15 |
+| P342 | matn-cov | [Covariance matrices of the matrix-normal distribution](/P/matn-cov) | JoramSoch | 2022-09-15 |
+| P343 | matn-marg | [Marginal distributions for the matrix-normal distribution](/P/matn-marg) | JoramSoch | 2022-09-15 |
diff --git a/I/PbT.md b/I/PbT.md
@@ -47,8 +47,10 @@ title: "Proof by Topic"
 - [Corrected Akaike information criterion in terms of maximum log-likelihood](/P/aicc-mll)
 - [Correlation always falls between -1 and +1](/P/corr-range)
 - [Correlation coefficient in terms of standard scores](/P/corr-z)
+- [Covariance matrices of the matrix-normal distribution](/P/matn-cov)
 - [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 of independent random variables](/P/cov-ind)
 - [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)
@@ -173,6 +175,7 @@ title: "Proof by Topic"
 
 ### M
 
+- [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)
 - [Marginal likelihood is a definite integral of joint likelihood](/P/ml-jl)
@@ -195,7 +198,9 @@ title: "Proof by Topic"
 - [Mean of the continuous uniform distribution](/P/cuni-mean)
 - [Mean of the exponential distribution](/P/exp-mean)
 - [Mean of the gamma distribution](/P/gam-mean)
+- [Mean of the matrix-normal distribution](/P/matn-mean)
 - [Mean of the multinomial distribution](/P/mult-mean)
+- [Mean of the multivariate normal distribution](/P/mvn-mean)
 - [Mean of the normal distribution](/P/norm-mean)
 - [Mean of the normal-gamma distribution](/P/ng-mean)
 - [Mean of the normal-Wishart distribution](/P/nw-mean)
@@ -337,7 +342,7 @@ title: "Proof by Topic"
 - [Relation of continuous mutual information to joint and conditional differential entropy](/P/cmi-jcde)
 - [Relation of continuous mutual information to marginal and conditional differential entropy](/P/cmi-mcde)
 - [Relation of continuous mutual information to marginal and joint differential entropy](/P/cmi-mjde)
-- [Relation of Kullback-Leibler divergence to entropy](/P/kl-ent)
+- [Relation of discrete Kullback-Leibler divergence to Shannon entropy](/P/kl-ent)
 - [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)
diff --git a/I/PwS.md b/I/PwS.md
@@ -46,6 +46,7 @@ title: "Proofs without Source"
 - [Log model evidence for the Poisson distribution with exposure values](/P/poissexp-lme)
 - [Log model evidence for the univariate Gaussian with known variance](/P/ugkv-lme)
 - [Log model evidence in terms of prior and posterior distribution](/P/lme-pnp)
+- [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)
 - [Marginal likelihood is a definite integral of joint likelihood](/P/ml-jl)
diff --git a/P/kl-ent.md b/P/kl-ent.md
@@ -7,7 +7,7 @@ affiliation: "BCCN Berlin"
 e_mail: "joram.soch@bccn-berlin.de"
 date: 2020-05-27 23:20:00
 
-title: "Relation of Kullback-Leibler divergence to entropy"
+title: "Relation of discrete Kullback-Leibler divergence to Shannon entropy"
 chapter: "General Theorems"
 section: "Information theory"
 topic: "Kullback-Leibler divergence"
diff --git a/P/matn-marg.md b/P/matn-marg.md
@@ -69,8 +69,8 @@ where $m_{ij}$ is the $(i,j)$-th entry of $M$.
 $$ \label{eq:A}
 A \in \mathbb{R}^{\lvert I \rvert \times n}, \quad \text{s.t.} \quad a_{ij} = \left\{
 \begin{array}{rl}
-0 \; , & \text{if} \; I_i = j \\
-1 \; , & \text{otherwise}
+1 \; , & \text{if} \; I_i = j \\
+0 \; , & \text{otherwise}
 \end{array}
 \right.
 $$
@@ -80,8 +80,8 @@ and define a selector matrix $B$, such that $b_{ij} = 1$, if the $j$-th column i
 $$ \label{eq:B}
 B \in \mathbb{R}^{p \times \lvert J \rvert}, \quad \text{s.t.} \quad b_{ij} = \left\{
 \begin{array}{rl}
-0 \; , & \text{if} \; J_j = i \\
-1 \; , & \text{otherwise} \; .
+1 \; , & \text{if} \; J_j = i \\
+0 \; , & \text{otherwise} \; .
 \end{array}
 \right.
 $$
diff --git a/P/matn-samp.md b/P/matn-samp.md
@@ -37,7 +37,7 @@ $$
 **Proof:** If all entries of $X$ are independent and [standard normally distributed](/D/snorm)
 
 $$ \label{eq:xij-dist}
-x_{ij} \sim \mathcal{N}(0, 1) \quad \text{ind. for all} \quad i = 1,\ldots,n \quad \text{and} \quad j = 1,\ldots,p \; ,
+x_{ij} \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0, 1) \quad \text{for all} \quad i = 1,\ldots,n \quad \text{and} \quad j = 1,\ldots,p \; ,
 $$
 
 this [implies a multivariate normal distribution with diagonal covariance matrix](/P/mvn-ind):
diff --git a/P/mult-cov.md b/P/mult-cov.md
@@ -46,13 +46,13 @@ $$
 which [has the variance](/P/bin-var) $\mathrm{Var}(X_i) = n p_i(1-p_i) = n (p_i - p_i^2)$, constituting the elements of the main diagonal in $\mathrm{Cov}(X)$ in \eqref{eq:mult-cov}. To prove $\mathrm{Cov}(X_i, X_j) = -n p_i p_j$ for $i \ne j$ (which constitutes the off-diagonal elements of the covariance matrix), we first recognize that
 
 $$ \label{eq:bin-sum}
-X_i = \sum_{k=1}^n \mathbb{I}_i(k), \quad \text{with} \quad \mathbb{I}_i(k) := \begin{cases}
+X_i = \sum_{k=1}^n \mathbb{I}_i(k), \quad \text{with} \quad \mathbb{I}_i(k) = \begin{cases}
     1 & \text{if $k$-th draw was of category $i$}, \\
-    0 & \text{otherwise},
+    0 & \text{otherwise} \; ,
 \end{cases}
 $$
 
-the indicator function $\mathbb{I}_i$ being a [Bernoulli-distributed](/D/bern) random variable with the [expected value](/P/bern-mean) $p_i$. Then, we have 
+where the indicator function $\mathbb{I}_i$ is a [Bernoulli-distributed](/D/bern) random variable with the [expected value](/P/bern-mean) $p_i$. Then, we have 
 
 $$ \label{eq:mult-cov-qed}
 \begin{split}
diff --git a/P/mvn-cov.md b/P/mvn-cov.md
@@ -102,7 +102,7 @@ $$ \label{eq:mvn-cov-qed}
 \begin{split}
 \mathrm{Cov}(x) &= \mathrm{Cov}( Az + \mu ) \\
 &\overset{\eqref{eq:cov-inv}}{=} \mathrm{Cov}(Az) \\
-&\overset{\eqref{eq:cov-scal}}{=} A \mathrm{Cov}(z) A^\mathrm{T} \\
+&\overset{\eqref{eq:cov-scal}}{=} A \, \mathrm{Cov}(z) A^\mathrm{T} \\
 &\overset{\eqref{eq:z-cov}}{=} A I_n A^\mathrm{T} \\
 &= A A^\mathrm{T} \\
 &= \Sigma \; .
diff --git a/P/mvn-mean.md b/P/mvn-mean.md
@@ -84,8 +84,8 @@ $$ \label{eq:mvn-mean-qed}
 \begin{split}
 \mathrm{E}(x) &= \mathrm{E}( Az + \mu ) \\
 &= \mathrm{E}(Az) + \mathrm{E}(\mu) \\
-&= A \mathrm{E}(z) + \mu \\
-&\overset{\eqref{eq:z-mean}}{=} A 0_n + \mu \\
+&= A \, \mathrm{E}(z) + \mu \\
+&\overset{\eqref{eq:z-mean}}{=} A \, 0_n + \mu \\
 &= \mu \; .
 \end{split}
 $$
diff --git a/P/ug-ttest1.md b/P/ug-ttest1.md
@@ -64,13 +64,13 @@ $$ \label{eq:var-samp}
 s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (y_i - \bar{y})^2 \; .
 $$
 
-Using the [linearity of the expected value](/P/mean-lin), the [additivity of the variance under independence](/P/var-add) and [scaling of the variance upon multiplication](/P/var-scal), the sample mean follows a [normal distribution](/D/norm)
+Using the [linear combination formula for normal random variables](/P/norm-lincomb), the sample mean follows a [normal distribution](/D/norm) with the following parameters:
 
 $$ \label{eq:mean-samp-dist}
-\bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i \sim \mathcal{N}\left( \frac{1}{n} n \mu, \left(\frac{1}{n}\right)^2 n \sigma^2 \right) = \mathcal{N}\left( \mu, \sigma^2/n \right)
+\bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i \sim \mathcal{N}\left( \frac{1}{n} n \mu, \left(\frac{1}{n}\right)^2 n \sigma^2 \right) = \mathcal{N}\left( \mu, \sigma^2/n \right) \; .
 $$
 
-and additionally using the [invariance of the variance under addition](/P/var-inv) and applying the null hypothesis from \eqref{eq:ttest1-h0}, the distribution of $Z = \sqrt{n}(\bar{y}-\mu_0)/\sigma$ becomes [standard normal](/D/snorm)
+Again employing the linear combination theorem and applying the null hypothesis from \eqref{eq:ttest1-h0}, the distribution of $Z = \sqrt{n}(\bar{y}-\mu_0)/\sigma$ becomes [standard normal](/D/snorm)
 
 $$ \label{eq:Z-dist}
 Z = \frac{\sqrt{n}(\bar{y}-\mu_0)}{\sigma} \sim \mathcal{N}\left( \frac{\sqrt{n}}{\sigma} (\mu - \mu_0), \left(\frac{\sqrt{n}}{\sigma}\right)^2 \frac{\sigma^2}{n} \right) \overset{H_0}{=} \mathcal{N}\left( 0, 1 \right) \; .
diff --git a/P/ug-ttest2.md b/P/ug-ttest2.md
@@ -42,7 +42,7 @@ y_{2i} &\sim \mathcal{N}(\mu_2, \sigma^2), \quad i = 1, \ldots, n_2
 \end{split}
 $$
 
-be a [univariate Gaussian data set](/D/ug) representing two groups of unequal size $n_1$ and $n_2$ with unknown means $\mu_1$ and $\mu_2$ and equal unknown variance $\sigma^2$. Then, the [test statistic](/D/tstat)
+be two [univariate Gaussian data sets](/D/ug) representing two groups of unequal size $n_1$ and $n_2$ with unknown means $\mu_1$ and $\mu_2$ and equal unknown variance $\sigma^2$. Then, the [test statistic](/D/tstat)
 
 $$ \label{eq:t}
 t = \frac{(\bar{y}_1-\bar{y}_2)-\mu_\Delta}{s_p \cdot \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}
@@ -85,16 +85,16 @@ s^2_2 &= \frac{1}{n_2-1} \sum_{i=1}^{n_2} (y_{2i} - \bar{y}_2)^2 \; .
 \end{split}
 $$
 
-Using the [linearity of the expected value](/P/mean-lin), the [additivity of the variance under independence](/P/var-add) and [scaling of the variance upon multiplication](/P/var-scal), the sample means follow a [normal distribution](/D/norm)
+Using the [linear combination formula for normal random variables](/P/norm-lincomb), the sample means follows [normal distributions](/D/norm) with the following parameters:
 
 $$ \label{eq:mean-samp-dist}
 \begin{split}
 \bar{y}_1 &= \frac{1}{n_1} \sum_{i=1}^{n_1} y_{1i} \sim \mathcal{N}\left( \frac{1}{n_1} n_1 \mu_1, \left(\frac{1}{n_1}\right)^2 n_1 \sigma^2 \right) = \mathcal{N}\left( \mu_1, \sigma^2/n_1 \right) \\
-\bar{y}_2 &= \frac{1}{n_2} \sum_{i=1}^{n_2} y_{2i} \sim \mathcal{N}\left( \frac{1}{n_2} n_2 \mu_2, \left(\frac{1}{n_2}\right)^2 n_2 \sigma^2 \right) = \mathcal{N}\left( \mu_2, \sigma^2/n_2 \right)
+\bar{y}_2 &= \frac{1}{n_2} \sum_{i=1}^{n_2} y_{2i} \sim \mathcal{N}\left( \frac{1}{n_2} n_2 \mu_2, \left(\frac{1}{n_2}\right)^2 n_2 \sigma^2 \right) = \mathcal{N}\left( \mu_2, \sigma^2/n_2 \right) \; .
 \end{split}
 $$
 
-and additionally using the [invariance of the variance under addition](/P/var-inv) and applying the null hypothesis from \eqref{eq:ttest2-h0}, the distribution of $Z = ( ( \bar{y}_1 - \bar{y}_2 ) - \mu_{\Delta} ) / ( \sigma \sqrt{1/n_1+1/n_2} )$ becomes [standard normal](/D/snorm)
+Again employing the linear combination theorem and applying the null hypothesis from \eqref{eq:ttest2-h0}, the distribution of $Z = ( ( \bar{y}_1 - \bar{y}_2 ) - \mu_{\Delta} ) / ( \sigma \sqrt{1/n_1+1/n_2} )$ becomes [standard normal](/D/snorm)
 
 $$ \label{eq:Z-dist}
 Z = \frac{(\bar{y}_1-\bar{y}_2)-\mu_\Delta}{\sigma \cdot \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}} \sim \mathcal{N}\left( \frac{(\mu_1-\mu_2)-\mu_\Delta}{\sigma \cdot \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}, \left(\frac{1}{\sigma \cdot \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\right)^2 \left( \frac{\sigma^2}{n_1} + \frac{\sigma^2}{n_2} \right) \right) \overset{H_0}{=} \mathcal{N}\left( 0, 1 \right) \; .
diff --git a/P/ugkv-ztest1.md b/P/ugkv-ztest1.md
@@ -64,13 +64,13 @@ $$ \label{eq:mean-samp}
 \bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i \; .
 $$
 
-Using the [linearity of the expected value](/P/mean-lin), the [additivity of the variance under independence](/P/var-add) and [scaling of the variance upon multiplication](/P/var-scal), the sample mean follows a [normal distribution](/D/norm)
+Using the [linear combination formula for normal random variables](/P/norm-lincomb), the sample mean follows a [normal distribution](/D/norm) with the following parameters:
 
 $$ \label{eq:mean-samp-dist}
-\bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i \sim \mathcal{N}\left( \frac{1}{n} n \mu, \left(\frac{1}{n}\right)^2 n \sigma^2 \right) = \mathcal{N}\left( \mu, \sigma^2/n \right)
+\bar{y} = \frac{1}{n} \sum_{i=1}^{n} y_i \sim \mathcal{N}\left( \frac{1}{n} n \mu, \left(\frac{1}{n}\right)^2 n \sigma^2 \right) = \mathcal{N}\left( \mu, \sigma^2/n \right) \; .
 $$
 
-and additionally using the [invariance of the variance under addition](/P/var-inv), the distribution of $z = \sqrt{n/\sigma^2} (\bar{y}-\mu_0)$ becomes
+Again employing the linear combination theorem, the distribution of $z = \sqrt{n/\sigma^2} (\bar{y}-\mu_0)$ becomes
 
 $$ \label{eq:z-dist-s1}
 z = \sqrt{\frac{n}{\sigma^2}} (\bar{y} - \mu_0) \sim \mathcal{N}\left( \sqrt{\frac{n}{\sigma^2}} (\mu - \mu_0), \left(\sqrt{\frac{n}{\sigma^2}}\right)^2 \frac{\sigma^2}{n} \right) = \mathcal{N}\left( \sqrt{n} \, \frac{\mu-\mu_0}{\sigma}, 1 \right) \; ,
diff --git a/P/ugkv-ztest2.md b/P/ugkv-ztest2.md
@@ -42,7 +42,7 @@ y_{2i} &\sim \mathcal{N}(\mu_2, \sigma_2^2), \quad i = 1, \ldots, n_2
 \end{split}
 $$
 
-be a [univariate Gaussian data set](/D/ug) representing two groups of unequal size $n_1$ and $n_2$ with unknown means $\mu_1$ and $\mu_2$ and unknown variances $\sigma_1^2$ and $\sigma_2^2$. Then, the [test statistic](/D/tstat)
+be two [univariate Gaussian data sets](/D/ug) representing two groups of unequal size $n_1$ and $n_2$ with unknown means $\mu_1$ and $\mu_2$ and unknown variances $\sigma_1^2$ and $\sigma_2^2$. Then, the [test statistic](/D/tstat)
 
 $$ \label{eq:z}
 z = \frac{(\bar{y}_1-\bar{y}_2)-\mu_\Delta}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}
@@ -70,16 +70,16 @@ $$ \label{eq:mean-samp}
 \end{split}
 $$
 
-Using the [linearity of the expected value](/P/mean-lin), the [additivity of the variance under independence](/P/var-add) and [scaling of the variance upon multiplication](/P/var-scal), the sample means follow a [normal distribution](/D/norm)
+Using the [linear combination formula for normal random variables](/P/norm-lincomb), the sample means follows [normal distributions](/D/norm) with the following parameters:
 
 $$ \label{eq:mean-samp-dist}
 \begin{split}
 \bar{y}_1 &= \frac{1}{n_1} \sum_{i=1}^{n_1} y_{1i} \sim \mathcal{N}\left( \frac{1}{n_1} n_1 \mu_1, \left(\frac{1}{n_1}\right)^2 n_1 \sigma^2 \right) = \mathcal{N}\left( \mu_1, \sigma_1^2/n_1 \right) \\
-\bar{y}_2 &= \frac{1}{n_2} \sum_{i=1}^{n_2} y_{2i} \sim \mathcal{N}\left( \frac{1}{n_2} n_2 \mu_2, \left(\frac{1}{n_2}\right)^2 n_2 \sigma^2 \right) = \mathcal{N}\left( \mu_2, \sigma_2^2/n_2 \right)
+\bar{y}_2 &= \frac{1}{n_2} \sum_{i=1}^{n_2} y_{2i} \sim \mathcal{N}\left( \frac{1}{n_2} n_2 \mu_2, \left(\frac{1}{n_2}\right)^2 n_2 \sigma^2 \right) = \mathcal{N}\left( \mu_2, \sigma_2^2/n_2 \right) \; .
 \end{split}
 $$
 
-and additionally using the [invariance of the variance under addition](/P/var-inv), the distribution of $z = [(\bar{y}_1-\bar{y}_2)-\mu_\Delta]/\sigma_\Delta$ becomes
+Again employing the linear combination theorem, the distribution of $z = [(\bar{y}_1-\bar{y}_2)-\mu_\Delta]/\sigma_\Delta$ becomes
 
 $$ \label{eq:z-dist-s1}
 z = \frac{(\bar{y}_1-\bar{y}_2)-\mu_\Delta}{\sigma_\Delta} \sim \mathcal{N}\left( \frac{(\mu_1-\mu_2)-\mu_\Delta}{\sigma_\Delta}, \left(\frac{1}{\sigma_\Delta}\right)^2 \sigma_\Delta^2 \right) = \mathcal{N}\left( \frac{(\mu_1-\mu_2)-\mu_\Delta}{\sigma_\Delta}, 1 \right)
diff --git a/S/ChI.md b/S/ChI.md
diff --git a/S/ChII.md b/S/ChII.md
diff --git a/S/ChIII.md b/S/ChIII.md
diff --git a/S/ChIV.md b/S/ChIV.md
diff --git a/_layouts/special.html b/_layouts/special.html
