Merge pull request #36 from StatProofBook/master

JoramSoch · web-flow · commit 81f648f140f1 · 2020-10-22T09:26:50.000+02:00
update to master
diff --git a/D/chi2.md b/D/chi2.md
@@ -45,7 +45,7 @@ $$\label{eq:chi2}
 Y = \sum_{i=1}^{k} X_{i}^{2} \sim \chi^{2}(k) \quad \text{where} \quad k > 0 \; .
 $$
 
-A [random variables](/D/rvar) $Y$ is said said to follow a chi-square distribution with $k$ number of degress of freedom, if and only if its [probability density function](/D/pdf) is given by
+A [random variable](/D/rvar) $Y$ is said said to follow a chi-square distribution with $k$ number of degress of freedom, if and only if its [probability density function](/D/pdf) is given by
 
 $$ \label{eq:chi2-pdf}
 \chi^{2}(x; k) = \frac{1}{2^{k/2}\Gamma (k/2)} \, x^{k/2-1} \, e^{-x/2}
diff --git a/D/sgam.md b/D/sgam.md
@@ -21,6 +21,13 @@ sources:
     pages: "retrieved on 2020-05-26"
     url: "https://github.com/JoramSoch/MACS/blob/master/MD_gamrnd.m"
     doi: "10.5281/zenodo.845404"
+  - authors: "NIST/SEMATECH"
+    year: 2012
+    title: "Gamma distribution"
+    in: "e-Handbook of Statistical Methods"
+    pages: "ch. 1.3.6.6.11"
+    url: "https://www.itl.nist.gov/div898/handbook/eda/section3/eda366b.htm"
+    doi: "10.18434/M32189"
 
 def_id: "D64"
 shortcut: "sgam"
diff --git a/I/Definition_by_Author.md b/I/Definition_by_Author.md
@@ -4,7 +4,7 @@ title: "Definition by Author"
 ---
 
 
-### JoramSoch (96 definitions)
+### JoramSoch (98 definitions)
 
 - [Akaike information criterion](/D/aic)
 - [Bayesian information criterion](/D/bic)
@@ -62,6 +62,8 @@ title: "Definition by Author"
 - [Matrix-normal distribution](/D/matn)
 - [Maximum likelihood estimation](/D/mle)
 - [Maximum log-likelihood](/D/mll)
+- [Median](/D/med)
+- [Mode](/D/mode)
 - [Moment](/D/mom)
 - [Moment-generating function](/D/mgf)
 - [Multinomial distribution](/D/mult)
diff --git a/I/Definition_by_Number.md b/I/Definition_by_Number.md
@@ -107,3 +107,5 @@ title: "Definition by Number"
 | D98 | mom-cent | [Central moment](/D/mom-cent) | JoramSoch | 2020-10-08 |
 | D99 | mom-stand | [Standardized moment](/D/mom-stand) | JoramSoch | 2020-10-08 |
 | D100 | chi2 | [Chi-square distribution](/D/chi2) | kjpetrykowski | 2020-10-13 |
+| D101 | med | [Median](/D/med) | JoramSoch | 2020-10-15 |
+| D102 | mode | [Mode](/D/mode) | JoramSoch | 2020-10-15 |
diff --git a/I/Definition_by_Topic.md b/I/Definition_by_Topic.md
@@ -96,6 +96,8 @@ title: "Definition by Topic"
 - [Matrix-normal distribution](/D/matn)
 - [Maximum likelihood estimation](/D/mle)
 - [Maximum log-likelihood](/D/mll)
+- [Median](/D/med)
+- [Mode](/D/mode)
 - [Moment](/D/mom)
 - [Moment-generating function](/D/mgf)
 - [Multinomial distribution](/D/mult)
diff --git a/I/Proof_by_Author.md b/I/Proof_by_Author.md
@@ -4,7 +4,7 @@ title: "Proof by Author"
 ---
 
 
-### JoramSoch (166 proofs)
+### JoramSoch (170 proofs)
 
 - [(Non-)Multiplicativity of the expected value](/P/mean-mult)
 - [Additivity of the Kullback-Leibler divergence for independent distributions](/P/kl-add)
@@ -28,6 +28,7 @@ title: "Proof by Author"
 - [Cumulative distribution function of the continuous uniform distribution](/P/cuni-cdf)
 - [Cumulative distribution function of the discrete uniform distribution](/P/duni-cdf)
 - [Cumulative distribution function of the exponential distribution](/P/exp-cdf)
+- [Cumulative distribution function of the gamma distribution](/P/gam-cdf)
 - [Cumulative distribution function of the normal distribution](/P/norm-cdf)
 - [Derivation of Bayesian model averaging](/P/bma-der)
 - [Derivation of R² and adjusted R²](/P/rsq-der)
@@ -42,6 +43,7 @@ title: "Proof by Author"
 - [Equivalence of matrix-normal distribution and multivariate normal distribution](/P/matn-mvn)
 - [Expectation of a quadratic form](/P/mean-qf)
 - [Expected value of a non-negative random variable](/P/mean-nnrvar)
+- [Expected value of x times ln(x) for a gamma distribution](/P/gam-xlogx)
 - [Exponential distribution is a special case of gamma distribution](/P/exp-gam)
 - [Expression of the cumulative distribution function of the normal distribution without the error function](/P/norm-cdfwerf)
 - [First central moment is zero](/P/momcent-1st)
@@ -144,7 +146,9 @@ title: "Proof by Author"
 - [Quantile function of the exponential distribution](/P/exp-qf)
 - [Quantile function of the normal distribution](/P/norm-qf)
 - [Relation between gamma distribution and standard gamma distribution](/P/gam-sgam)
+- [Relation between gamma distribution and standard gamma distribution](/P/gam-sgam2)
 - [Relation between normal distribution and standard normal distribution](/P/norm-snorm)
+- [Relation between normal distribution and standard normal distribution](/P/norm-snorm2)
 - [Relation of Kullback-Leibler divergence to entropy](/P/kl-ent)
 - [Relation of continuous Kullback-Leibler divergence to differential entropy](/P/kl-dent)
 - [Relation of continuous mutual information to joint and conditional differential entropy](/P/cmi-jcde)
diff --git a/I/Proof_by_Number.md b/I/Proof_by_Number.md
@@ -182,3 +182,7 @@ title: "Proof by Number"
 | P173 | momcent-2nd | [Second central moment is variance](/P/momcent-2nd) | JoramSoch | 2020-10-08 |
 | P174 | chi2-gam | [Chi-square distribution is a special case of gamma distribution](/P/chi2-gam) | kjpetrykowski | 2020-10-12 |
 | P175 | chi2-mom | [Moments of the chi-square distribution](/P/chi2-mom) | kjpetrykowski | 2020-10-13 |
+| P176 | norm-snorm2 | [Relation between normal distribution and standard normal distribution](/P/norm-snorm2) | JoramSoch | 2020-10-15 |
+| P177 | gam-sgam2 | [Relation between gamma distribution and standard gamma distribution](/P/gam-sgam2) | JoramSoch | 2020-10-15 |
+| P178 | gam-cdf | [Cumulative distribution function of the gamma distribution](/P/gam-cdf) | JoramSoch | 2020-10-15 |
+| P179 | gam-xlogx | [Expected value of x times ln(x) for a gamma distribution](/P/gam-xlogx) | JoramSoch | 2020-10-15 |
diff --git a/I/Proof_by_Topic.md b/I/Proof_by_Topic.md
@@ -38,6 +38,7 @@ title: "Proof by Topic"
 - [Cumulative distribution function of the continuous uniform distribution](/P/cuni-cdf)
 - [Cumulative distribution function of the discrete uniform distribution](/P/duni-cdf)
 - [Cumulative distribution function of the exponential distribution](/P/exp-cdf)
+- [Cumulative distribution function of the gamma distribution](/P/gam-cdf)
 - [Cumulative distribution function of the normal distribution](/P/norm-cdf)
 
 ### D
@@ -59,6 +60,7 @@ title: "Proof by Topic"
 - [Equivalence of matrix-normal distribution and multivariate normal distribution](/P/matn-mvn)
 - [Expectation of a quadratic form](/P/mean-qf)
 - [Expected value of a non-negative random variable](/P/mean-nnrvar)
+- [Expected value of x times ln(x) for a gamma distribution](/P/gam-xlogx)
 - [Exponential distribution is a special case of gamma distribution](/P/exp-gam)
 - [Expression of the cumulative distribution function of the normal distribution without the error function](/P/norm-cdfwerf)
 
@@ -202,7 +204,9 @@ title: "Proof by Topic"
 ### R
 
 - [Relation between gamma distribution and standard gamma distribution](/P/gam-sgam)
+- [Relation between gamma distribution and standard gamma distribution](/P/gam-sgam2)
 - [Relation between normal distribution and standard normal distribution](/P/norm-snorm)
+- [Relation between normal distribution and standard normal distribution](/P/norm-snorm2)
 - [Relation of Kullback-Leibler divergence to entropy](/P/kl-ent)
 - [Relation of continuous Kullback-Leibler divergence to differential entropy](/P/kl-dent)
 - [Relation of continuous mutual information to joint and conditional differential entropy](/P/cmi-jcde)
diff --git a/I/Proofs_without_Source.md b/I/Proofs_without_Source.md
@@ -9,6 +9,7 @@ title: "Proofs without Source"
 - [Cumulative distribution function of the continuous uniform distribution](/P/cuni-cdf)
 - [Cumulative distribution function of the discrete uniform distribution](/P/duni-cdf)
 - [Cumulative distribution function of the exponential distribution](/P/exp-cdf)
+- [Cumulative distribution function of the gamma distribution](/P/gam-cdf)
 - [Derivation of Bayesian model averaging](/P/bma-der)
 - [Derivation of the log Bayes factor](/P/lbf-der)
 - [Derivation of the log family evidence](/P/lfe-der)
@@ -58,7 +59,9 @@ title: "Proofs without Source"
 - [Quantile function of the discrete uniform distribution](/P/duni-qf)
 - [Quantile function of the exponential distribution](/P/exp-qf)
 - [Relation between gamma distribution and standard gamma distribution](/P/gam-sgam)
+- [Relation between gamma distribution and standard gamma distribution](/P/gam-sgam2)
 - [Relation between normal distribution and standard normal distribution](/P/norm-snorm)
+- [Relation between normal distribution and standard normal distribution](/P/norm-snorm2)
 - [Relationship between R² and maximum log-likelihood](/P/rsq-mll)
 - [Relationship between covariance and correlation](/P/cov-corr)
 - [Relationship between precision matrix and correlation matrix](/P/precmat-corrmat)
diff --git a/P/cuni-mean.md b/P/cuni-mean.md
@@ -37,7 +37,7 @@ $$
 **Proof:** The [expected value](/D/mean) is the probability-weighted average over all possible values:
 
 $$ \label{eq:mean}
-\mathrm{E}(X) = \int_{\mathbb{R}} x \cdot f_X(x) \, \mathrm{d}x \; .
+\mathrm{E}(X) = \int_{\mathcal{X}} x \cdot f_X(x) \, \mathrm{d}x \; .
 $$
 
 With the [probability density function of the continuous uniform distribution](/P/cuni-pdf), this becomes:
diff --git a/P/exp-mean.md b/P/exp-mean.md
@@ -44,7 +44,7 @@ $$
 **Proof:** The [expected value](/D/mean) is the probability-weighted average over all possible values:
 
 $$ \label{eq:mean}
-\mathrm{E}(X) = \int_{\mathbb{R}} x \cdot f_\mathrm{X}(x) \, \mathrm{d}x \; .
+\mathrm{E}(X) = \int_{\mathcal{X}} x \cdot f_\mathrm{X}(x) \, \mathrm{d}x \; .
 $$
 
 With the [probability density function of the exponential distribution](/P/exp-pdf), this reads:
diff --git a/P/gam-logmean.md b/P/gam-logmean.md
@@ -36,9 +36,11 @@ $$
 Then, the [expectation](/D/mean) of the natural logarithm of $X$ is
 
 $$ \label{eq:gam-logmean}
-\mathrm{E}(\ln X) = \psi(a) - \ln(b) \; .
+\mathrm{E}(\ln X) = \psi(a) - \ln(b)
 $$
 
+where $\psi(x)$ is the digamma function.
+
 
 **Proof:** Let $Y = \ln(X)$, such that $\mathrm{E}(Y) = \mathrm{E}(\ln X)$ and consider the special case that $b = 1$. In this case, the [probability density function of the gamma distribution](/P/gam-pdf) is
 
diff --git a/P/gam-mean.md b/P/gam-mean.md
@@ -43,7 +43,7 @@ $$
 **Proof:** The [expected value](/D/mean) is the probability-weighted average over all possible values:
 
 $$ \label{eq:mean}
-\mathrm{E}(X) = \int_{\mathbb{R}} x \cdot f_X(x) \, \mathrm{d}x \; .
+\mathrm{E}(X) = \int_{\mathcal{X}} x \cdot f_X(x) \, \mathrm{d}x \; .
 $$
 
 With the [probability density function of the gamma distribution](/P/gam-pdf), this reads:
diff --git a/P/gam-sgam.md b/P/gam-sgam.md
@@ -27,17 +27,17 @@ $$ \label{eq:X-gam}
 X \sim \mathrm{Gam}(a,b) \; .
 $$
 
-Then, the quantity $Z = b X$ will have a [standard gamma distribution](/D/sgam) with shape $a$ and rate $1$:
+Then, the quantity $Y = b X$ will have a [standard gamma distribution](/D/sgam) with shape $a$ and rate $1$:
 
-$$ \label{eq:Z-snorm}
-Z = b X \sim \mathrm{Gam}(a,1) \; .
+$$ \label{eq:Y-snorm}
+Y = b X \sim \mathrm{Gam}(a,1) \; .
 $$
 
 
-**Proof:** Rearranging to get $X$ in terms of $Z$, we have
+**Proof:** Rearranging to get $X$ in terms of $Y$, we have
 
-$$ \label{eq:X-Z}
-X = \frac{1}{b} Z \; .
+$$ \label{eq:X-Y}
+X = \frac{1}{b} Y \; .
 $$
 
 The [cumulative distribution function of the gamma-distributed](/P/gam-cdf) $X$ is
@@ -46,13 +46,13 @@ $$ \label{eq:gam-cdf}
 F_X(t) = \int_{-\infty}^{t} \frac{b^a}{\Gamma(a)} x^{a-1} \exp[-b x] \, \mathrm{d}x \; .
 $$
 
-Substituting \eqref{eq:X-Z} into \eqref{eq:gam-cdf}, we obtain
+Substituting \eqref{eq:X-Y} into \eqref{eq:gam-cdf}, we obtain
 
 $$ \label{eq:sgam-cdf}
 \begin{split}
-F_Z(t) &= \int_{-\infty}^{t} \frac{b^a}{\Gamma(a)} \left(\frac{1}{b} z\right)^{a-1} \exp\left[-b \left(\frac{1}{b} z\right)\right] \, \mathrm{d}\left(\frac{1}{b} z\right) \\
-&= \int_{-\infty}^{t} \frac{b^a}{b} \left(\frac{1}{b}\right)^{a-1} \cdot \frac{1}{\Gamma(a)} z^{a-1} \exp[-z] \, \mathrm{d}z \\
-&= \int_{-\infty}^{t} \frac{1}{\Gamma(a)} z^{a-1} \exp[-z] \, \mathrm{d}z
+F_Z(t) &= \int_{-\infty}^{t} \frac{b^a}{\Gamma(a)} \left(\frac{1}{b} y\right)^{a-1} \exp\left[-b \left(\frac{1}{b} y\right)\right] \, \mathrm{d}\left(\frac{1}{b} y\right) \\
+&= \int_{-\infty}^{t} \frac{b^a}{b} \left(\frac{1}{b}\right)^{a-1} \cdot \frac{1}{\Gamma(a)} y^{a-1} \exp[-y] \, \mathrm{d}y \\
+&= \int_{-\infty}^{t} \frac{1}{\Gamma(a)} y^{a-1} \exp[-y] \, \mathrm{d}y
 \end{split}
 $$
 
diff --git a/P/gam-xlogx.md b/P/gam-xlogx.md
@@ -40,7 +40,7 @@ $$ \label{eq:gam-xlogx}
 $$
 
 
-**Proof:** With the [definition of the expected value](/D/mean) and the [probability density function of the gamma distribution](/P/gam-pdf), we have:
+**Proof:** With the [definition of the expected value](/D/mean), the [law of the unconscious statistician](/P/mean-lotus) and the [probability density function of the gamma distribution](/P/gam-pdf), we have:
 
 $$ \label{eq:gam-xlogx-s1}
 \begin{split}
@@ -68,7 +68,7 @@ $$ \label{eq:gam-fct}
 \Gamma(x+1) = \Gamma(x) \cdot x \quad \Leftrightarrow \quad \frac{\Gamma(x+1)}{\Gamma(x)} = x \; ,
 $$
 
-the expression in equation \eqref{eq:gam-mean-s1} develops into:
+the expression in equation \eqref{eq:gam-xlogx-s1} develops into:
 
 $$ \label{eq:gam-xlogx-qed}
 \mathrm{E}(X \ln X) = \frac{a}{b} \left[ \psi(a) - \ln(b) \right] \; .
diff --git a/P/norm-mean.md b/P/norm-mean.md
@@ -43,7 +43,7 @@ $$
 **Proof:** The [expected value](/D/mean) is the probability-weighted average over all possible values:
 
 $$ \label{eq:mean}
-\mathrm{E}(X) = \int_{\mathbb{R}} x \cdot f_X(x) \, \mathrm{d}x \; .
+\mathrm{E}(X) = \int_{\mathcal{X}} x \cdot f_X(x) \, \mathrm{d}x \; .
 $$
 
 With the [probability density function of the normal distribution](/P/norm-pdf), this reads:
