From de755ad7339e7ee5b3caba9ef1b4f3a56c6eeb95 Mon Sep 17 00:00:00 2001
From: 0PrashantYadav0 <devprashantkyadav@gmail.com>
Date: Sun, 18 Oct 2026 09:35:43 +0530
Subject: [PATCH] docs: fix README equations in `stats/base/dists/beta` and
 `betaprime`

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---
---
 .../@stdlib/stats/base/dists/beta/logpdf/README.md          | 6 +++---
 .../@stdlib/stats/base/dists/beta/pdf/README.md             | 6 +++---
 .../@stdlib/stats/base/dists/beta/skewness/README.md        | 6 +++---
 .../@stdlib/stats/base/dists/beta/stdev/README.md           | 6 +++---
 .../@stdlib/stats/base/dists/betaprime/logpdf/README.md     | 6 +++---
 .../@stdlib/stats/base/dists/betaprime/pdf/README.md        | 6 +++---
 6 files changed, 18 insertions(+), 18 deletions(-)

diff --git a/lib/node_modules/@stdlib/stats/base/dists/beta/logpdf/README.md b/lib/node_modules/@stdlib/stats/base/dists/beta/logpdf/README.md
index fb19196d0040..c386525f5b05 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/beta/logpdf/README.md
+++ b/lib/node_modules/@stdlib/stats/base/dists/beta/logpdf/README.md
@@ -26,13 +26,13 @@ limitations under the License.
 
 The [probability density function][pdf] (PDF) for a [beta][beta-distribution] random variable is
 
-<!-- <equation class="equation" label="eq:beta_pdf" align="center" raw="f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) + \Gamma(\beta)}{x^{\alpha-1}(1-x)^{\beta-1}} & \text{ for } x \in (0,1) \\ 0 & \text{ otherwise } \end{cases}" alt="Probability density function (PDF) for a beta distribution."> -->
+<!-- <equation class="equation" label="eq:beta_pdf" align="center" raw="f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}{x^{\alpha-1}(1-x)^{\beta-1}} & \text{ for } x \in (0,1) \\ 0 & \text{ otherwise } \end{cases}" alt="Probability density function (PDF) for a beta distribution."> -->
 
 ```math
-f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) + \Gamma(\beta)}{x^{\alpha-1}(1-x)^{\beta-1}} & \text{ for } x \in (0,1) \\ 0 & \text{ otherwise } \end{cases}
+f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}{x^{\alpha-1}(1-x)^{\beta-1}} & \text{ for } x \in (0,1) \\ 0 & \text{ otherwise } \end{cases}
 ```
 
-<!-- <div class="equation" align="center" data-raw-text="f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) + \Gamma(\beta)}{x^{\alpha-1}(1-x)^{\beta-1}} &amp; \text{ for } x \in (0,1) \\ 0 &amp; \text{ otherwise } \end{cases}" data-equation="eq:beta_pdf">
+<!-- <div class="equation" align="center" data-raw-text="f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}{x^{\alpha-1}(1-x)^{\beta-1}} &amp; \text{ for } x \in (0,1) \\ 0 &amp; \text{ otherwise } \end{cases}" data-equation="eq:beta_pdf">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/beta/logpdf/docs/img/equation_beta_pdf.svg" alt="Probability density function (PDF) for a beta distribution.">
     <br>
 </div> -->
diff --git a/lib/node_modules/@stdlib/stats/base/dists/beta/pdf/README.md b/lib/node_modules/@stdlib/stats/base/dists/beta/pdf/README.md
index c9861d5d4756..d950b621a9ce 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/beta/pdf/README.md
+++ b/lib/node_modules/@stdlib/stats/base/dists/beta/pdf/README.md
@@ -26,13 +26,13 @@ limitations under the License.
 
 The [probability density function][pdf] (PDF) for a [beta][beta-distribution] random variable is
 
-<!-- <equation class="equation" label="eq:beta_pdf" align="center" raw="f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) + \Gamma(\beta)}{x^{\alpha-1}(1-x)^{\beta-1}} & \text{ for } x \in (0,1) \\ 0 & \text{ otherwise } \end{cases}" alt="Probability density function (PDF) for a beta distribution."> -->
+<!-- <equation class="equation" label="eq:beta_pdf" align="center" raw="f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}{x^{\alpha-1}(1-x)^{\beta-1}} & \text{ for } x \in (0,1) \\ 0 & \text{ otherwise } \end{cases}" alt="Probability density function (PDF) for a beta distribution."> -->
 
 ```math
-f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) + \Gamma(\beta)}{x^{\alpha-1}(1-x)^{\beta-1}} & \text{ for } x \in (0,1) \\ 0 & \text{ otherwise } \end{cases}
+f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}{x^{\alpha-1}(1-x)^{\beta-1}} & \text{ for } x \in (0,1) \\ 0 & \text{ otherwise } \end{cases}
 ```
 
-<!-- <div class="equation" align="center" data-raw-text="f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) + \Gamma(\beta)}{x^{\alpha-1}(1-x)^{\beta-1}} &amp; \text{ for } x \in (0,1) \\ 0 &amp; \text{ otherwise } \end{cases}" data-equation="eq:beta_pdf">
+<!-- <div class="equation" align="center" data-raw-text="f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}{x^{\alpha-1}(1-x)^{\beta-1}} &amp; \text{ for } x \in (0,1) \\ 0 &amp; \text{ otherwise } \end{cases}" data-equation="eq:beta_pdf">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/beta/pdf/docs/img/equation_beta_pdf.svg" alt="Probability density function (PDF) for a beta distribution.">
     <br>
 </div> -->
diff --git a/lib/node_modules/@stdlib/stats/base/dists/beta/skewness/README.md b/lib/node_modules/@stdlib/stats/base/dists/beta/skewness/README.md
index 72aeaacbd88b..1f144395c0e0 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/beta/skewness/README.md
+++ b/lib/node_modules/@stdlib/stats/base/dists/beta/skewness/README.md
@@ -28,13 +28,13 @@ limitations under the License.
 
 The [skewness][skewness] for a [beta][beta-distribution] random variable is
 
-<!-- <equation class="equation" label="eq:beta_skewness" align="center" raw="\operatorname{skew}\left( X \right) = \frac{1-2p}{\sqrt{np(1-p)}}" alt="Skewness for a beta distribution."> -->
+<!-- <equation class="equation" label="eq:beta_skewness" align="center" raw="\operatorname{skew}\left( X \right) = \frac{2(\beta-\alpha)\sqrt{\alpha+\beta+1}}{(\alpha+\beta+2)\sqrt{\alpha\beta}}" alt="Skewness for a beta distribution."> -->
 
 ```math
-\mathop{\mathrm{skew}}\left( X \right) = \frac{1-2p}{\sqrt{np(1-p)}}
+\mathop{\mathrm{skew}}\left( X \right) = \frac{2(\beta-\alpha)\sqrt{\alpha+\beta+1}}{(\alpha+\beta+2)\sqrt{\alpha\beta}}
 ```
 
-<!-- <div class="equation" align="center" data-raw-text="\operatorname{skew}\left( X \right) = \frac{1-2p}{\sqrt{np(1-p)}}" data-equation="eq:beta_skewness">
+<!-- <div class="equation" align="center" data-raw-text="\operatorname{skew}\left( X \right) = \frac{2(\beta-\alpha)\sqrt{\alpha+\beta+1}}{(\alpha+\beta+2)\sqrt{\alpha\beta}}" data-equation="eq:beta_skewness">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/beta/skewness/docs/img/equation_beta_skewness.svg" alt="Skewness for a beta distribution.">
     <br>
 </div> -->
diff --git a/lib/node_modules/@stdlib/stats/base/dists/beta/stdev/README.md b/lib/node_modules/@stdlib/stats/base/dists/beta/stdev/README.md
index 3c87c97f60ab..7c2253012b8c 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/beta/stdev/README.md
+++ b/lib/node_modules/@stdlib/stats/base/dists/beta/stdev/README.md
@@ -28,13 +28,13 @@ limitations under the License.
 
 The [standard deviation][stdev] for a [beta][beta-distribution] random variable is
 
-<!-- <equation class="equation" label="eq:beta_stdev" align="center" raw="\sigma = (\alpha+\beta) \sqrt{\tfrac{\alpha\beta}{\alpha+\beta+1}}" alt="Standard deviation for a beta distribution."> -->
+<!-- <equation class="equation" label="eq:beta_stdev" align="center" raw="\sigma = \frac{\sqrt{\alpha\beta}}{(\alpha+\beta)\sqrt{\alpha+\beta+1}}" alt="Standard deviation for a beta distribution."> -->
 
 ```math
-\sigma = (\alpha+\beta) \sqrt{\tfrac{\alpha\beta}{\alpha+\beta+1}}
+\sigma = \frac{\sqrt{\alpha\beta}}{(\alpha+\beta)\sqrt{\alpha+\beta+1}}
 ```
 
-<!-- <div class="equation" align="center" data-raw-text="\sigma = (\alpha+\beta) \sqrt{\tfrac{\alpha\beta}{\alpha+\beta+1}}" data-equation="eq:beta_stdev">
+<!-- <div class="equation" align="center" data-raw-text="\sigma = \frac{\sqrt{\alpha\beta}}{(\alpha+\beta)\sqrt{\alpha+\beta+1}}" data-equation="eq:beta_stdev">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/beta/stdev/docs/img/equation_beta_stdev.svg" alt="Standard deviation for a beta distribution.">
     <br>
 </div> -->
diff --git a/lib/node_modules/@stdlib/stats/base/dists/betaprime/logpdf/README.md b/lib/node_modules/@stdlib/stats/base/dists/betaprime/logpdf/README.md
index 36b5a80a1b1e..b1a62ec2f562 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/betaprime/logpdf/README.md
+++ b/lib/node_modules/@stdlib/stats/base/dists/betaprime/logpdf/README.md
@@ -26,13 +26,13 @@ limitations under the License.
 
 The [probability density function][pdf] (PDF) for a [beta prime][betaprime-distribution] random variable is
 
-<!-- <equation class="equation" label="eq:betaprime_pdf" align="center" raw="f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) + \Gamma(\beta)}{x^{\alpha-1}(1+x)^{-\alpha-\beta}} & \text{ for } x > 0 \\ 0 & \text{ otherwise } \end{cases}" alt="Probability density function (PDF) for a beta prime distribution."> -->
+<!-- <equation class="equation" label="eq:betaprime_pdf" align="center" raw="f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}{x^{\alpha-1}(1+x)^{-\alpha-\beta}} & \text{ for } x > 0 \\ 0 & \text{ otherwise } \end{cases}" alt="Probability density function (PDF) for a beta prime distribution."> -->
 
 ```math
-f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) + \Gamma(\beta)}{x^{\alpha-1}(1+x)^{-\alpha-\beta}} & \text{ for } x > 0 \\ 0 & \text{ otherwise } \end{cases}
+f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}{x^{\alpha-1}(1+x)^{-\alpha-\beta}} & \text{ for } x > 0 \\ 0 & \text{ otherwise } \end{cases}
 ```
 
-<!-- <div class="equation" align="center" data-raw-text="f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) + \Gamma(\beta)}{x^{\alpha-1}(1+x)^{-\alpha-\beta}} &amp; \text{ for } x &gt; 0 \\ 0 &amp; \text{ otherwise } \end{cases}" data-equation="eq:betaprime_pdf">
+<!-- <div class="equation" align="center" data-raw-text="f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}{x^{\alpha-1}(1+x)^{-\alpha-\beta}} &amp; \text{ for } x &gt; 0 \\ 0 &amp; \text{ otherwise } \end{cases}" data-equation="eq:betaprime_pdf">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/betaprime/logpdf/docs/img/equation_betaprime_pdf.svg" alt="Probability density function (PDF) for a beta prime distribution.">
     <br>
 </div> -->
diff --git a/lib/node_modules/@stdlib/stats/base/dists/betaprime/pdf/README.md b/lib/node_modules/@stdlib/stats/base/dists/betaprime/pdf/README.md
index ec6158fff99b..45b7956066e3 100644
--- a/lib/node_modules/@stdlib/stats/base/dists/betaprime/pdf/README.md
+++ b/lib/node_modules/@stdlib/stats/base/dists/betaprime/pdf/README.md
@@ -26,13 +26,13 @@ limitations under the License.
 
 The [probability density function][pdf] (PDF) for a [beta prime][betaprime-distribution] random variable is
 
-<!-- <equation class="equation" label="eq:betaprime_pdf" align="center" raw="f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) + \Gamma(\beta)}{x^{\alpha-1}(1+x)^{-\alpha-\beta}} & \text{ for } x > 0 \\ 0 & \text{ otherwise } \end{cases}" alt="Probability density function (PDF) for a beta prime distribution."> -->
+<!-- <equation class="equation" label="eq:betaprime_pdf" align="center" raw="f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}{x^{\alpha-1}(1+x)^{-\alpha-\beta}} & \text{ for } x > 0 \\ 0 & \text{ otherwise } \end{cases}" alt="Probability density function (PDF) for a beta prime distribution."> -->
 
 ```math
-f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) + \Gamma(\beta)}{x^{\alpha-1}(1+x)^{-\alpha-\beta}} & \text{ for } x > 0 \\ 0 & \text{ otherwise } \end{cases}
+f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}{x^{\alpha-1}(1+x)^{-\alpha-\beta}} & \text{ for } x > 0 \\ 0 & \text{ otherwise } \end{cases}
 ```
 
-<!-- <div class="equation" align="center" data-raw-text="f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) + \Gamma(\beta)}{x^{\alpha-1}(1+x)^{-\alpha-\beta}} &amp; \text{ for } x &gt; 0 \\ 0 &amp; \text{ otherwise } \end{cases}" data-equation="eq:betaprime_pdf">
+<!-- <div class="equation" align="center" data-raw-text="f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}{x^{\alpha-1}(1+x)^{-\alpha-\beta}} &amp; \text{ for } x &gt; 0 \\ 0 &amp; \text{ otherwise } \end{cases}" data-equation="eq:betaprime_pdf">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/betaprime/pdf/docs/img/equation_betaprime_pdf.svg" alt="Probability density function (PDF) for a beta prime distribution.">
     <br>
 </div> -->