Minor fixes to beta function parameters and clarifying text for gamma distribution

src/functions-reference/positive_continuous_distributions.qmd

@@ -360,11 +360,13 @@ For a description of argument and return types, see section
 
 ### Probability density function
 
-If $\alpha \in \mathbb{R}^+$ and $\beta \in \mathbb{R}^+$, then for $y
+If the shape parameter $\alpha \in \mathbb{R}^+$ and the rate (or inverse scale) parameter $\beta \in \mathbb{R}^+$, then for $y
 \in \mathbb{R}^+$, \begin{equation*} \text{Gamma}(y|\alpha,\beta) =
 \frac{\beta^{\alpha}}      {\Gamma(\alpha)} \, y^{\alpha - 1}
 \exp(-\beta \, y) . \end{equation*}
 
+Under the shape and rate formulation of the Gamma distribution, $\mathbb{E}[y] = \alpha / \beta$ and $\textrm{var}[y] = \alpha / \beta^2$.
+
 ### Distribution statement
 
 `y ~ ` **`gamma`**`(alpha, beta)`

src/functions-reference/real-valued_basic_functions.qmd

@@ -1193,7 +1193,7 @@ beta. The beta function, $\text{B}(\alpha,\beta)$, computes the
 normalizing constant for the beta distribution, and is defined for
 $\alpha > 0$ and $\beta > 0$.
 \begin{equation*}
-\text{lbeta}(\alpha,\beta) = \log \Gamma(a) + \log \Gamma(b) - \log \Gamma(a+b)
+\text{lbeta}(\alpha,\beta) = \log \Gamma(\alpha) + \log \Gamma(\beta) - \log \Gamma(\alpha+\beta)
 \end{equation*}
 See section [appendix](mathematical_functions.qmd#beta-appendix) for definition of $\text{B}(\alpha, \beta)$.
 {{< since 2.0 >}}