Merge pull request #155 from StatProofBook/master

JoramSoch · web-flow · commit 3842ff822548 · 2024-11-01T14:32:56.000+01:00
update to master
diff --git a/D/bvn.md b/D/bvn.md
@@ -10,8 +10,8 @@ date: 2023-09-22 10:56:27
 title: "Bivariate normal distribution"
 chapter: "Probability Distributions"
 section: "Multivariate continuous distributions"
-topic: "Multivariate normal distribution"
-definition: "Bivariate normal distribution"
+topic: "Bivariate normal distribution"
+definition: "Definition"
 
 sources:
   - authors: "Wikipedia"
diff --git a/D/dist-uni.md b/D/dist-uni.md
@@ -11,7 +11,7 @@ title: "Unimodal and multimodal probability distribution"
 chapter: "General Theorems"
 section: "Probability theory"
 topic: "Probability distributions"
-definition: "Unimodal vs. multimodal"
+definition: "Unimodal vs. multimodal distribution"
 
 sources:
   - authors: "Weisstein, Eric W."
diff --git a/Gemfile.lock b/Gemfile.lock
@@ -59,16 +59,14 @@ GEM
     rb-fsevent (0.10.3)
     rb-inotify (0.10.0)
       ffi (~> 1.0)
-    rexml (3.3.6)
-      strscan
+    rexml (3.3.9)
     rouge (3.9.0)
     ruby_dep (1.5.0)
     safe_yaml (1.0.5)
     sassc (2.1.0)
       ffi (~> 1.9)
     sassc (2.1.0-x64-mingw32)
       ffi (~> 1.9)
-    strscan (3.1.0)
     terminal-table (1.8.0)
       unicode-display_width (~> 1.1, >= 1.1.1)
     thread_safe (0.3.6)
diff --git a/I/DbA.md b/I/DbA.md
@@ -13,7 +13,7 @@ title: "Definition by Author"
 
 - [Sample covariance](/D/cov-samp)
 
-### JoramSoch (190 definitions)
+### JoramSoch (191 definitions)
 
 - [Akaike information criterion](/D/aic)
 - [Alternative hypothesis](/D/h1)
@@ -198,6 +198,7 @@ title: "Definition by Author"
 - [Two-way analysis of variance](/D/anova2)
 - [Uniform and non-uniform prior distribution](/D/prior-uni)
 - [Uniform-prior log model evidence](/D/uplme)
+- [Unimodal and multimodal probability distribution](/D/dist-uni)
 - [Univariate and multivariate random variable](/D/rvar-uni)
 - [Univariate Gaussian](/D/ug)
 - [Univariate Gaussian with known variance](/D/ugkv)
diff --git a/I/DbN.md b/I/DbN.md
@@ -213,3 +213,4 @@ title: "Definition by Number"
 | D204 | para | [Parameter](/D/para) | JoramSoch | 2024-09-27 |
 | D205 | stat | [Statistic](/D/stat) | JoramSoch | 2024-10-04 |
 | D206 | dof | [Degrees of freedom](/D/dof) | JoramSoch | 2024-10-11 |
+| D207 | dist-uni | [Unimodal and multimodal probability distribution](/D/dist-uni) | JoramSoch | 2024-10-25 |
diff --git a/I/DbT.md b/I/DbT.md
@@ -262,6 +262,7 @@ title: "Definition by Topic"
 
 - [Uniform and non-uniform prior distribution](/D/prior-uni)
 - [Uniform-prior log model evidence](/D/uplme)
+- [Unimodal and multimodal probability distribution](/D/dist-uni)
 - [Univariate and multivariate random variable](/D/rvar-uni)
 - [Univariate Gaussian](/D/ug)
 - [Univariate Gaussian with known variance](/D/ugkv)
diff --git a/I/PbA.md b/I/PbA.md
@@ -12,7 +12,7 @@ title: "Proof by Author"
 
 - [Posterior predictive distribution is a marginal distribution of the joint likelihood](/P/postpred-jl)
 
-### JoramSoch (438 proofs)
+### JoramSoch (439 proofs)
 
 - [Accuracy and complexity for Bayesian linear regression](/P/blr-anc)
 - [Accuracy and complexity for Bayesian linear regression with known covariance](/P/blrkc-anc)
@@ -171,6 +171,7 @@ title: "Proof by Author"
 - [Law of total expectation](/P/mean-tot)
 - [Law of total probability](/P/prob-tot)
 - [Law of total variance](/P/var-tot)
+- [Linear combination of bivariate normal random variables](/P/bvn-lincomb)
 - [Linear combination of independent normal random variables](/P/norm-lincomb)
 - [Linear transformation theorem for the matrix-normal distribution](/P/matn-ltt)
 - [Linear transformation theorem for the moment-generating function](/P/mgf-ltt)
diff --git a/I/PbN.md b/I/PbN.md
@@ -480,3 +480,4 @@ title: "Proof by Number"
 | P472 | corr-ind | [Independent random variables are uncorrelated](/P/corr-ind) | JoramSoch | 2024-09-27 |
 | P473 | norm-corrind | [Normally distributed and uncorrelated does not imply independent](/P/norm-corrind) | JoramSoch | 2024-10-04 |
 | P474 | norm-margjoint | [Marginally normal does not imply jointly normal](/P/norm-margjoint) | JoramSoch | 2024-10-11 |
+| P475 | bvn-lincomb | [Linear combination of bivariate normal random variables](/P/bvn-lincomb) | JoramSoch | 2024-10-25 |
diff --git a/I/PbT.md b/I/PbT.md
@@ -198,6 +198,7 @@ title: "Proof by Topic"
 - [Law of total expectation](/P/mean-tot)
 - [Law of total probability](/P/prob-tot)
 - [Law of total variance](/P/var-tot)
+- [Linear combination of bivariate normal random variables](/P/bvn-lincomb)
 - [Linear combination of independent normal random variables](/P/norm-lincomb)
 - [Linear transformation theorem for the matrix-normal distribution](/P/matn-ltt)
 - [Linear transformation theorem for the moment-generating function](/P/mgf-ltt)
diff --git a/I/ToC.md b/I/ToC.md
@@ -589,47 +589,50 @@ title: "Table of Contents"
    &emsp;&ensp; 4.1.1. *[Definition](/D/mvn)* <br>
    &emsp;&ensp; 4.1.2. **[Special case of matrix-normal distribution](/P/mvn-matn)** <br>
    &emsp;&ensp; 4.1.3. **[Relationship to chi-squared distribution](/P/mvn-chi2)** <br>
-   &emsp;&ensp; 4.1.4. *[Bivariate normal distribution](/D/bvn)* <br>
-   &emsp;&ensp; 4.1.5. **[Probability density function of the bivariate normal distribution](/P/bvn-pdf)** <br>
-   &emsp;&ensp; 4.1.6. **[Probability density function in terms of correlation coefficient](/P/bvn-pdfcorr)** <br>
-   &emsp;&ensp; 4.1.7. **[Linear combination of bivariate normal random variables](/P/bvn-lincomb)** <br>
-   &emsp;&ensp; 4.1.8. **[Probability density function](/P/mvn-pdf)** <br>
-   &emsp;&ensp; 4.1.9. **[Moment-generating function](/P/mvn-mgf)** <br>
-   &emsp;&ensp; 4.1.10. **[Mean](/P/mvn-mean)** <br>
-   &emsp;&ensp; 4.1.11. **[Covariance](/P/mvn-cov)** <br>
-   &emsp;&ensp; 4.1.12. **[Differential entropy](/P/mvn-dent)** <br>
-   &emsp;&ensp; 4.1.13. **[Kullback-Leibler divergence](/P/mvn-kl)** <br>
-   &emsp;&ensp; 4.1.14. **[Linear transformation](/P/mvn-ltt)** <br>
-   &emsp;&ensp; 4.1.15. **[Marginal distributions](/P/mvn-marg)** <br>
-   &emsp;&ensp; 4.1.16. **[Conditional distributions](/P/mvn-cond)** <br>
-   &emsp;&ensp; 4.1.17. **[Conditions for independence](/P/mvn-ind)** <br>
-   &emsp;&ensp; 4.1.18. **[Independence of products](/P/mvn-indprod)** <br>
+   &emsp;&ensp; 4.1.4. **[Probability density function](/P/mvn-pdf)** <br>
+   &emsp;&ensp; 4.1.5. **[Moment-generating function](/P/mvn-mgf)** <br>
+   &emsp;&ensp; 4.1.6. **[Mean](/P/mvn-mean)** <br>
+   &emsp;&ensp; 4.1.7. **[Covariance](/P/mvn-cov)** <br>
+   &emsp;&ensp; 4.1.8. **[Differential entropy](/P/mvn-dent)** <br>
+   &emsp;&ensp; 4.1.9. **[Kullback-Leibler divergence](/P/mvn-kl)** <br>
+   &emsp;&ensp; 4.1.10. **[Linear transformation](/P/mvn-ltt)** <br>
+   &emsp;&ensp; 4.1.11. **[Marginal distributions](/P/mvn-marg)** <br>
+   &emsp;&ensp; 4.1.12. **[Conditional distributions](/P/mvn-cond)** <br>
+   &emsp;&ensp; 4.1.13. **[Conditions for independence](/P/mvn-ind)** <br>
+   &emsp;&ensp; 4.1.14. **[Independence of products](/P/mvn-indprod)** <br>
+   
+   <p id="Bivariate normal distribution"></p>
+   4.2. Bivariate normal distribution <br>
+   &emsp;&ensp; 4.2.1. *[Definition](/D/bvn)* <br>
+   &emsp;&ensp; 4.2.2. **[Probability density function](/P/bvn-pdf)** <br>
+   &emsp;&ensp; 4.2.3. **[Probability density function in terms of correlation coefficient](/P/bvn-pdfcorr)** <br>
+   &emsp;&ensp; 4.2.4. **[Linear combination](/P/bvn-lincomb)** <br>
    
    <p id="Multivariate t-distribution"></p>
-   4.2. Multivariate t-distribution <br>
-   &emsp;&ensp; 4.2.1. *[Definition](/D/mvt)* <br>
-   &emsp;&ensp; 4.2.2. **[Probability density function](/P/mvt-pdf)** <br>
-   &emsp;&ensp; 4.2.3. **[Relationship to F-distribution](/P/mvt-f)** <br>
+   4.3. Multivariate t-distribution <br>
+   &emsp;&ensp; 4.3.1. *[Definition](/D/mvt)* <br>
+   &emsp;&ensp; 4.3.2. **[Probability density function](/P/mvt-pdf)** <br>
+   &emsp;&ensp; 4.3.3. **[Relationship to F-distribution](/P/mvt-f)** <br>
    
    <p id="Normal-gamma distribution"></p>
-   4.3. Normal-gamma distribution <br>
-   &emsp;&ensp; 4.3.1. *[Definition](/D/ng)* <br>
-   &emsp;&ensp; 4.3.2. **[Special case of normal-Wishart distribution](/P/ng-nw)** <br>
-   &emsp;&ensp; 4.3.3. **[Probability density function](/P/ng-pdf)** <br>
-   &emsp;&ensp; 4.3.4. **[Mean](/P/ng-mean)** <br>
-   &emsp;&ensp; 4.3.5. **[Covariance](/P/ng-cov)** <br>
-   &emsp;&ensp; 4.3.6. **[Differential entropy](/P/ng-dent)** <br>
-   &emsp;&ensp; 4.3.7. **[Kullback-Leibler divergence](/P/ng-kl)** <br>
-   &emsp;&ensp; 4.3.8. **[Marginal distributions](/P/ng-marg)** <br>
-   &emsp;&ensp; 4.3.9. **[Conditional distributions](/P/ng-cond)** <br>
-   &emsp;&ensp; 4.3.10. **[Drawing samples](/P/ng-samp)** <br>
+   4.4. Normal-gamma distribution <br>
+   &emsp;&ensp; 4.4.1. *[Definition](/D/ng)* <br>
+   &emsp;&ensp; 4.4.2. **[Special case of normal-Wishart distribution](/P/ng-nw)** <br>
+   &emsp;&ensp; 4.4.3. **[Probability density function](/P/ng-pdf)** <br>
+   &emsp;&ensp; 4.4.4. **[Mean](/P/ng-mean)** <br>
+   &emsp;&ensp; 4.4.5. **[Covariance](/P/ng-cov)** <br>
+   &emsp;&ensp; 4.4.6. **[Differential entropy](/P/ng-dent)** <br>
+   &emsp;&ensp; 4.4.7. **[Kullback-Leibler divergence](/P/ng-kl)** <br>
+   &emsp;&ensp; 4.4.8. **[Marginal distributions](/P/ng-marg)** <br>
+   &emsp;&ensp; 4.4.9. **[Conditional distributions](/P/ng-cond)** <br>
+   &emsp;&ensp; 4.4.10. **[Drawing samples](/P/ng-samp)** <br>
    
    <p id="Dirichlet distribution"></p>
-   4.4. Dirichlet distribution <br>
-   &emsp;&ensp; 4.4.1. *[Definition](/D/dir)* <br>
-   &emsp;&ensp; 4.4.2. **[Probability density function](/P/dir-pdf)** <br>
-   &emsp;&ensp; 4.4.3. **[Kullback-Leibler divergence](/P/dir-kl)** <br>
-   &emsp;&ensp; 4.4.4. **[Exceedance probabilities](/P/dir-ep)** <br>
+   4.5. Dirichlet distribution <br>
+   &emsp;&ensp; 4.5.1. *[Definition](/D/dir)* <br>
+   &emsp;&ensp; 4.5.2. **[Probability density function](/P/dir-pdf)** <br>
+   &emsp;&ensp; 4.5.3. **[Kullback-Leibler divergence](/P/dir-kl)** <br>
+   &emsp;&ensp; 4.5.4. **[Exceedance probabilities](/P/dir-ep)** <br>
 
 5. <p id="Matrix-variate continuous distributions">Matrix-variate continuous distributions</p>
    
diff --git a/P/bin-test.md b/P/bin-test.md
@@ -88,7 +88,7 @@ $$ \label{eq:y-cvals}
 \begin{split}
 \alpha &\geq \mathrm{Pr}(y \in \left\lbrace 0, \ldots, y_1 \right\rbrace \cup \left\lbrace y_2, \ldots, n \right\rbrace \vert H_0) \\
 &= \mathrm{Pr}(y \leq y_1 \vert H_0) + \mathrm{Pr}(y \geq y_2 \vert H_0) \\
-&= \mathrm{Pr}(y \leq y_1 \vert H_0) + (1-\mathrm{Pr}(y \leq (y_2-1) \vert H_0) \; .
+&= \mathrm{Pr}(y \leq y_1 \vert H_0) + \left( 1 - \mathrm{Pr}(y \leq (y_2-1) \vert H_0) \right) \; .
 \end{split}
 $$
 
diff --git a/P/bvn-lincomb.md b/P/bvn-lincomb.md
@@ -10,8 +10,8 @@ date: 2024-10-25 11:59:37
 title: "Linear combination of bivariate normal random variables"
 chapter: "Probability Distributions"
 section: "Multivariate continuous distributions"
-topic: "Multivariate normal distribution"
-theorem: "Linear combination of bivariate normal random variables"
+topic: "Bivariate normal distribution"
+theorem: "Linear combination"
 
 sources:
   - authors: "Wikipedia"
diff --git a/P/bvn-pdf.md b/P/bvn-pdf.md
@@ -10,8 +10,8 @@ date: 2023-09-22 12:59:11
 title: "Probability density function of the bivariate normal distribution"
 chapter: "Probability Distributions"
 section: "Multivariate continuous distributions"
-topic: "Multivariate normal distribution"
-theorem: "Probability density function of the bivariate normal distribution"
+topic: "Bivariate normal distribution"
+theorem: "Probability density function"
 
 sources:
 
diff --git a/P/bvn-pdfcorr.md b/P/bvn-pdfcorr.md
@@ -10,7 +10,7 @@ date: 2023-09-29 09:45:10
 title: "Probability density function of the bivariate normal distribution in terms of correlation coefficient"
 chapter: "Probability Distributions"
 section: "Multivariate continuous distributions"
-topic: "Multivariate normal distribution"
+topic: "Bivariate normal distribution"
 theorem: "Probability density function in terms of correlation coefficient"
 
 sources:
diff --git a/P/prob-exc.md b/P/prob-exc.md
@@ -40,7 +40,7 @@ $$ \label{eq:exc}
 p(A,B) = 0 \; .
 $$
 
-The [addition law of probability](/D/prob-marg) states that
+The [addition law of probability](/D/prob-add) states that
 
 $$ \label{eq:prob-add-set}
 p(A \cup B) = p(A) + p(B) - p(A \cap B)
diff --git a/P/ugkv-ztest2.md b/P/ugkv-ztest2.md
@@ -79,7 +79,7 @@ $$ \label{eq:mean-samp-dist}
 \end{split}
 $$
 
-Again employing the linear combination theorem, the distribution of $z = [(\bar{y}_1-\bar{y}_2)-\mu_\Delta]/\sigma_\Delta$ becomes
+Again employing the linear combination theorem, the distribution of $z = \left[ (\bar{y}\_1-\bar{y}\_2) - \mu_{\Delta} \right] / \sigma_{\Delta}$ 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)
