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docs: fix README equations in stats/base/dists/beta and betaprime #16222

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6 changes: 3 additions & 3 deletions lib/node_modules/@stdlib/stats/base/dists/beta/logpdf/README.md
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Expand Up @@ -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> -->
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6 changes: 3 additions & 3 deletions lib/node_modules/@stdlib/stats/base/dists/beta/pdf/README.md
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Expand Up @@ -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> -->
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6 changes: 3 additions & 3 deletions lib/node_modules/@stdlib/stats/base/dists/beta/skewness/README.md
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Expand Up @@ -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> -->
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6 changes: 3 additions & 3 deletions lib/node_modules/@stdlib/stats/base/dists/beta/stdev/README.md
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Expand Up @@ -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> -->
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6 changes: 3 additions & 3 deletions lib/node_modules/@stdlib/stats/base/dists/betaprime/logpdf/README.md
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Expand Up @@ -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> -->
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6 changes: 3 additions & 3 deletions lib/node_modules/@stdlib/stats/base/dists/betaprime/pdf/README.md
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Expand Up @@ -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> -->
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