updated index pages

JoramSoch · JoramSoch · commit bd98f8330646 · 2025-10-24T17:49:59.000+02:00
The index pages were updated with the recently added proofs.
diff --git a/I/PbA.md b/I/PbA.md
@@ -16,7 +16,7 @@ title: "Proof by Author"
 
 - [Posterior predictive distribution is a marginal distribution of the joint likelihood](/P/postpred-jl)
 
-### JoramSoch (480 proofs)
+### JoramSoch (483 proofs)
 
 - [Accuracy and complexity for Bayesian linear regression](/P/blr-anc)
 - [Accuracy and complexity for Bayesian linear regression with known covariance](/P/blrkc-anc)
@@ -267,11 +267,14 @@ title: "Proof by Author"
 - [Mean of the normal-gamma distribution](/P/ng-mean)
 - [Mean of the normal-Wishart distribution](/P/nw-mean)
 - [Mean of the Poisson distribution](/P/poiss-mean)
+- [Median of the beta distribution](/P/beta-med)
 - [Median of the continuous uniform distribution](/P/cuni-med)
 - [Median of the exponential distribution](/P/exp-med)
+- [Median of the gamma distribution](/P/gam-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 beta distribution](/P/beta-mode)
 - [Mode of the continuous uniform distribution](/P/cuni-mode)
 - [Mode of the gamma distribution](/P/gam-mode)
 - [Mode of the multivariate normal distribution](/P/mvn-mode)
@@ -511,7 +514,7 @@ title: "Proof by Author"
 ### kjpetrykowski (2 proofs)
 
 - [Chi-squared distribution is a special case of gamma distribution](/P/chi2-gam)
-- [Moments of the chi-squared distribution](/P/chi2-mom)
+- [Raw moments of the chi-squared distribution](/P/chi2-momraw)
 
 ### majapavlo (8 proofs)
 
diff --git a/I/PbN.md b/I/PbN.md
@@ -180,7 +180,7 @@ title: "Proof by Number"
 | P172 | momraw-2nd | [Relationship between second raw moment, variance and mean](/P/momraw-2nd) | JoramSoch | 2020-10-08 |
 | P173 | momcent-2nd | [Second central moment is variance](/P/momcent-2nd) | JoramSoch | 2020-10-08 |
 | P174 | chi2-gam | [Chi-squared distribution is a special case of gamma distribution](/P/chi2-gam) | kjpetrykowski | 2020-10-12 |
-| P175 | chi2-mom | [Moments of the chi-squared distribution](/P/chi2-mom) | kjpetrykowski | 2020-10-13 |
+| P175 | chi2-momraw | [Raw moments of the chi-squared distribution](/P/chi2-momraw) | kjpetrykowski | 2020-10-13 |
 | P176 | norm-snorm2 | [Relationship between normal distribution and standard normal distribution](/P/norm-snorm2) | JoramSoch | 2020-10-15 |
 | P177 | gam-sgam2 | [Relationship 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 |
@@ -523,3 +523,6 @@ title: "Proof by Number"
 | P515 | kl-nonneg3 | [Non-negativity of the Kullback-Leibler divergence](/P/kl-nonneg3) | JoramSoch | 2025-09-25 |
 | P516 | fren-dec | [Decompositions of the variational free energy](/P/fren-dec) | JoramSoch | 2025-09-25 |
 | P517 | fren-lme | [The variational free energy is a lower bound on the log model evidence](/P/fren-lme) | JoramSoch | 2025-09-25 |
+| P518 | gam-med | [Median of the gamma distribution](/P/gam-med) | JoramSoch | 2025-10-24 |
+| P519 | beta-med | [Median of the beta distribution](/P/beta-med) | JoramSoch | 2025-10-24 |
+| P520 | beta-mode | [Mode of the beta distribution](/P/beta-mode) | JoramSoch | 2025-10-24 |
diff --git a/I/PbT.md b/I/PbT.md
@@ -296,14 +296,17 @@ title: "Proof by Topic"
 - [Mean of the normal-Wishart distribution](/P/nw-mean)
 - [Mean of the Poisson distribution](/P/poiss-mean)
 - [Mean of the Wald distribution](/P/wald-mean)
+- [Median of the beta distribution](/P/beta-med)
 - [Median of the continuous uniform distribution](/P/cuni-med)
 - [Median of the exponential distribution](/P/exp-med)
+- [Median of the gamma distribution](/P/gam-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)
 - [Method of moments for ex-Gaussian-distributed data](/P/exg-mome)
 - [Method of moments for Wald-distributed data](/P/wald-mome)
+- [Mode of the beta distribution](/P/beta-mode)
 - [Mode of the continuous uniform distribution](/P/cuni-mode)
 - [Mode of the exponential distribution](/P/exp-mode)
 - [Mode of the gamma distribution](/P/gam-mode)
@@ -321,7 +324,6 @@ title: "Proof by Topic"
 - [Moment-generating function of the multivariate normal distribution](/P/mvn-mgf)
 - [Moment-generating function of the normal distribution](/P/norm-mgf)
 - [Moment-generating function of the Wald distribution](/P/wald-mgf)
-- [Moments of the chi-squared distribution](/P/chi2-mom)
 - [Monotonicity of probability](/P/prob-mon)
 - [Monotonicity of probability](/P/prob-mon2)
 - [Monotonicity of the expected value](/P/mean-mono)
@@ -471,6 +473,7 @@ title: "Proof by Topic"
 - [Range of probability](/P/prob-range)
 - [Range of the variance of the Bernoulli distribution](/P/bern-varrange)
 - [Range of the variance of the binomial distribution](/P/bin-varrange)
+- [Raw moments of the chi-squared distribution](/P/chi2-momraw)
 - [Redundancy of parameters describing the matrix-normal distribution](/P/matn-red)
 - [Redundancy of parameters describing the matrix-normal distribution](/P/matn-red2)
 - [Relation of continuous Kullback-Leibler divergence to differential entropy](/P/kl-dent)
diff --git a/I/ToC.md b/I/ToC.md
@@ -586,7 +586,7 @@ title: "Table of Contents"
    &emsp;&ensp; 3.7.1. *[Definition](/D/chi2)* <br>
    &emsp;&ensp; 3.7.2. **[Special case of gamma distribution](/P/chi2-gam)** <br>
    &emsp;&ensp; 3.7.3. **[Probability density function](/P/chi2-pdf)** <br>
-   &emsp;&ensp; 3.7.4. **[Moments](/P/chi2-mom)** <br>
+   &emsp;&ensp; 3.7.4. **[Raw moments](/P/chi2-momraw)** <br>
    
    <p id="F-distribution"></p>
    3.8. F-distribution <br>
diff --git a/P/beta-mode.md b/P/beta-mode.md
@@ -225,4 +225,4 @@ Thus, $f'\'\_X(x)$ is negative for $x = \frac{\alpha-1}{\alpha+\beta-2}$, demons
 
 * If $\alpha = 1$ and $\beta = 1$, then $f_X(x)$ is constant and any value in the interval $\left[ 0,1 \right]$ can be seen as the mode of $X$.
 
-* If $\alpha \geq 1$ and $\beta \geq 1$ (but not $\alpha = 1$ and $\beta = 1$), then $0 < x =  < 1$ and $f'_X(x) = 0$ and $f'\'\_X(x) < 0$, such that $f_X(x)$ reaches its machimum at $\mathrm{mode}(X) = \frac{\alpha-1}{\alpha+\beta-2}$.
+* If $\alpha \geq 1$ and $\beta \geq 1$ (but not $\alpha = 1$ and $\beta = 1$), then $0 < x = \frac{\alpha-1}{\alpha+\beta-2} < 1$ and $f'_X(x) = 0$ and $f'\'\_X(x) < 0$, such that $f_X(x)$ reaches its machimum at $\frac{\alpha-1}{\alpha+\beta-2} = \mathrm{mode}(X)$.
diff --git a/S/ChII.md b/S/ChII.md
@@ -25,7 +25,7 @@ shortcut: "ChII"
 | Gamma distribution | *[gam](/D/gam)*<br>*[sgam](/D/sgam)* | **[gam-pdf](/P/gam-pdf)** | **[gam-cdf](/P/gam-cdf)** | **[gam-qf](/P/gam-qf)** |  | **[gam-mean](/P/gam-mean)**<br>**[gam-logmean](/P/gam-logmean)**<br>**[gam-xlogx](/P/gam-xlogx)** |  |  | **[gam-var](/P/gam-var)** | **[gam-dent](/P/gam-dent)** | **[gam-kl](/P/gam-kl)** |  |  | **[gam-sgam](/P/gam-sgam)**<br>**[gam-wish](/P/gam-wish)**<br>**[gam-scal](/P/gam-scal)** |
 | Exponential distribution | *[exp](/D/exp)* | **[exp-pdf](/P/exp-pdf)** | **[exp-cdf](/P/exp-cdf)** | **[exp-qf](/P/exp-qf)** | **[exp-mgf](/P/exp-mgf)** | **[exp-mean](/P/exp-mean)** | **[exp-med](/P/exp-med)** | **[exp-mode](/P/exp-mode)** | **[exp-var](/P/exp-var)** |  |  |  |  | **[exp-gam](/P/exp-gam)**<br>**[exp-skew](/P/exp-skew)** |
 | Log-normal distribution | *[lognorm](/D/lognorm)* | **[lognorm-pdf](/P/lognorm-pdf)** | **[lognorm-cdf](/P/lognorm-cdf)** | **[lognorm-qf](/P/lognorm-qf)** |  | **[lognorm-mean](/P/lognorm-mean)** | **[lognorm-med](/P/lognorm-med)** | **[lognorm-mode](/P/lognorm-mode)** | **[lognorm-var](/P/lognorm-var)** |  |  |  |  |  |
-| Chi-squared distribution | *[chi2](/D/chi2)* | **[chi2-pdf](/P/chi2-pdf)** |  |  |  |  |  |  |  |  |  |  |  | **[chi2-gam](/P/chi2-gam)**<br>**[chi2-mom](/P/chi2-mom)** |
+| Chi-squared distribution | *[chi2](/D/chi2)* | **[chi2-pdf](/P/chi2-pdf)** |  |  |  |  |  |  |  |  |  |  |  | **[chi2-gam](/P/chi2-gam)**<br>**[chi2-momraw](/P/chi2-momraw)** |
 | F-distribution | *[f](/D/f)* | **[f-pdf](/P/f-pdf)** |  |  |  |  |  |  |  |  |  |  |  |  |
 | Beta distribution | *[beta](/D/beta)* | **[beta-pdf](/P/beta-pdf)** | **[beta-cdf](/P/beta-cdf)** |  | **[beta-mgf](/P/beta-mgf)** | **[beta-mean](/P/beta-mean)** |  |  | **[beta-var](/P/beta-var)** |  |  |  |  | **[beta-chi2](/P/beta-chi2)** |
 | Wald distribution | *[wald](/D/wald)* | **[wald-pdf](/P/wald-pdf)** |  |  | **[wald-mgf](/P/wald-mgf)** | **[wald-mean](/P/wald-mean)** |  |  | **[wald-var](/P/wald-var)** |  |  |  |  | **[wald-skew](/P/wald-skew)**<br>**[wald-mome](/P/wald-mome)** |

Original file line number	Diff line number	Diff line change
`@@ -225,4 +225,4 @@ Thus, $f'\'\_X(x)$ is negative for $x = \frac{\alpha-1}{\alpha+\beta-2}$, demons`
`225`	`225`
`226`	`226`	`* If $\alpha = 1$ and $\beta = 1$, then $f_X(x)$ is constant and any value in the interval $\left[ 0,1 \right]$ can be seen as the mode of $X$.`
`227`	`227`
`228`		`-* If $\alpha \geq 1$ and $\beta \geq 1$ (but not $\alpha = 1$ and $\beta = 1$), then $0 < x = < 1$ and $f'_X(x) = 0$ and $f'\'\_X(x) < 0$, such that $f_X(x)$ reaches its machimum at $\mathrm{mode}(X) = \frac{\alpha-1}{\alpha+\beta-2}$.`
	`228`	`+* If $\alpha \geq 1$ and $\beta \geq 1$ (but not $\alpha = 1$ and $\beta = 1$), then $0 < x = \frac{\alpha-1}{\alpha+\beta-2} < 1$ and $f'_X(x) = 0$ and $f'\'\_X(x) < 0$, such that $f_X(x)$ reaches its machimum at $\frac{\alpha-1}{\alpha+\beta-2} = \mathrm{mode}(X)$.`