Merge pull request #66 from StatProofBook/master

update to master

Author: JoramSoch

D/cf.md

@@ -20,6 +20,12 @@ sources:
     in: "Wikipedia, the free encyclopedia"
     pages: "retrieved on 2021-09-22"
     url: "https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory)#Definition"
+  - authors: "Taboga, Marco"
+    year: 2017
+    title: "Joint characteristic function"
+    in: "Lectures on probability and mathematical statistics"
+    pages: "retrieved on 2021-10-07"
+    url: "https://www.statlect.com/fundamentals-of-probability/joint-characteristic-function"
 
 def_id: "D159"
 shortcut: "cf"

D/mgf.md

@@ -20,6 +20,12 @@ sources:
     in: "Wikipedia, the free encyclopedia"
     pages: "retrieved on 2020-01-22"
     url: "https://en.wikipedia.org/wiki/Moment-generating_function#Definition"
+  - authors: "Taboga, Marco"
+    year: 2017
+    title: "Joint moment generating function"
+    in: "Lectures on probability and mathematical statistics"
+    pages: "retrieved on 2021-10-07"
+    url: "https://www.statlect.com/fundamentals-of-probability/joint-moment-generating-function"
 
 def_id: "D2"
 shortcut: "mgf"

I/Proof_by_Author.md

@@ -4,13 +4,14 @@ title: "Proof by Author"
 ---
 
 
-### JoramSoch (250 proofs)
+### JoramSoch (252 proofs)
 
 - [(Non-)Multiplicativity of the expected value](/P/mean-mult)
 - [Accuracy and complexity for the univariate Gaussian](/P/ug-anc)
 - [Accuracy and complexity for the univariate Gaussian with known variance](/P/ugkv-anc)
 - [Addition law of probability](/P/prob-add)
 - [Addition of the differential entropy upon multiplication with a constant](/P/dent-add)
+- [Addition of the differential entropy upon multiplication with invertible matrix](/P/dent-addvec)
 - [Additivity of the Kullback-Leibler divergence for independent distributions](/P/kl-add)
 - [Additivity of the variance for independent random variables](/P/var-add)
 - [Bayes' rule](/P/bayes-rule)
@@ -142,6 +143,7 @@ title: "Proof by Author"
 - [Monotonicity of probability](/P/prob-mon)
 - [Monotonicity of the expected value](/P/mean-mono)
 - [Necessary and sufficient condition for independence of multivariate normal random variables](/P/mvn-ind)
+- [Non-invariance of the differential entropy under change of variables](/P/dent-noninv)
 - [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)

I/Proof_by_Number.md

@@ -266,3 +266,5 @@ title: "Proof by Number"
 | P258 | pdf-sumind | [Probability density function of a sum of independent discrete random variables](/P/pdf-sumind) | JoramSoch | 2021-08-30 |
 | P259 | cf-fct | [Characteristic function of a function of a random variable](/P/cf-fct) | JoramSoch | 2021-09-22 |
 | P260 | mgf-fct | [Moment-generating function of a function of a random variable](/P/mgf-fct) | JoramSoch | 2021-09-22 |
+| P261 | dent-addvec | [Addition of the differential entropy upon multiplication with invertible matrix](/P/dent-addvec) | JoramSoch | 2021-10-07 |
+| P262 | dent-noninv | [Non-invariance of the differential entropy under change of variables](/P/dent-noninv) | JoramSoch | 2021-10-07 |

I/Proof_by_Topic.md

@@ -14,6 +14,7 @@ title: "Proof by Topic"
 - [Accuracy and complexity for the univariate Gaussian with known variance](/P/ugkv-anc)
 - [Addition law of probability](/P/prob-add)
 - [Addition of the differential entropy upon multiplication with a constant](/P/dent-add)
+- [Addition of the differential entropy upon multiplication with invertible matrix](/P/dent-addvec)
 - [Additivity of the Kullback-Leibler divergence for independent distributions](/P/kl-add)
 - [Additivity of the variance for independent random variables](/P/var-add)
 
@@ -186,6 +187,7 @@ title: "Proof by Topic"
 ### N
 
 - [Necessary and sufficient condition for independence of multivariate normal random variables](/P/mvn-ind)
+- [Non-invariance of the differential entropy under change of variables](/P/dent-noninv)
 - [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)

P/dent-addvec.md

@@ -86,7 +86,7 @@ $$ \label{eq:Y-dent-s2}
 \end{split}
 $$
 
-Using the differential $\mathrm{d}(Ax) = \left|A\right| \mathrm{d}x$, this becomes
+Using the differential $\mathrm{d}(Ax) = \lvert A \rvert \mathrm{d}x$, this becomes
 
 $$ \label{eq:Y-dent-s3}
 \begin{split}

P/dent-noninv.md

@@ -119,7 +119,7 @@ $$ \label{eq:Y-dent-s2}
 \end{split}
 $$
 
-Using the relations $y = f(x) \Rightarrow \mathrm{d}y = \left| J_f(x) \right| \, \mathrm{d}x$ and $\left|A\right|\left|B\right| = \left|AB\right|$, this becomes
+Using the relations $y = f(x) \Rightarrow \mathrm{d}y = \lvert J_f(x) \rvert \, \mathrm{d}x$ and $\lvert A \rvert \lvert B \rvert = \lvert AB \rvert$, this becomes
 
 $$ \label{eq:Y-dent-s3}
 \begin{split}
@@ -128,7 +128,7 @@ $$ \label{eq:Y-dent-s3}
 \end{split}
 $$
 
-Finally, employing [the fact](/D/pdf) that $\int_{\mathcal{X}} f_X(x) \, \mathrm{d}x = 1$ and the determinant property $\left|A^{-1}\right| = 1/\left|A\right|$, we can derive the [differential entropy](/D/dent) of $Y$ as
+Finally, employing [the fact](/D/pdf) that $\int_{\mathcal{X}} f_X(x) \, \mathrm{d}x = 1$ and the determinant property $\lvert A^{-1} \rvert = 1/\lvert A \rvert$, we can derive the [differential entropy](/D/dent) of $Y$ as
 
 $$ \label{eq:Y-dent-s4}
 \begin{split}

P/pdf-invfct.md

@@ -109,7 +109,7 @@ $$ \label{eq:A-y}
 A(y) = \left\lbrace x \in \mathbb{R}^n: g_j(x) \leq y_j \; \text{for all} \; j = 1, \ldots, n \right\rbrace
 $$
 
-and $g_j(X)$ is the function which returns the $j$-th element of $Y$, given a vector $x$.
+and $g_j(X)$ is the function which returns the $j$-th element of $Y$, given a vector $X$.
 
 <br>
 2) Next, we substitute $x = g^{-1}(y)$ into the integral which gives us