punctuation

Author: Solomon Kurz

09.Rmd

@@ -130,15 +130,15 @@ d %>%
 
 Behold the probability density for the generalized normal distribution:
 
-$$\text{Pr} (y | \mu, \alpha, \beta) = \frac{\beta}{2 \alpha \Gamma \bigg (\frac{1}{\beta} \bigg )} e ^ {- \bigg (\frac{|y - \mu|}{\alpha} \bigg ) ^ {\beta}}$$
+$$\text{Pr} (y | \mu, \alpha, \beta) = \frac{\beta}{2 \alpha \Gamma \bigg (\frac{1}{\beta} \bigg )} e ^ {- \bigg (\frac{|y - \mu|}{\alpha} \bigg ) ^ {\beta}},$$
 
-In this formulation, $\alpha =$ the scale, $\beta =$ the shape, $\mu =$ the location, and $\Gamma =$ the [gamma function](https://en.wikipedia.org/wiki/Gamma_function). If you read closely in the text, you'll discover that the densities in the right panel of Figure 9.2 were all created with the constraint $\sigma^2 = 1$. But $\sigma^2 \neq \alpha$ and there's no $\sigma$ in the equations in the text. However, it appears the variance for the generalized normal distribution follows the form
+where $\alpha =$ the scale, $\beta =$ the shape, $\mu =$ the location, and $\Gamma =$ the [gamma function](https://en.wikipedia.org/wiki/Gamma_function). If you read closely in the text, you'll discover that the densities in the right panel of Figure 9.2 were all created with the constraint $\sigma^2 = 1$. But $\sigma^2 \neq \alpha$ and there's no $\sigma$ in the equations in the text. However, it appears the variance for the generalized normal distribution follows the form
 
 $$\sigma^2 = \frac{\alpha^2 \Gamma (3/\beta)}{\Gamma (1/\beta)}.$$
 
-So if you do the algebra, you'll see that you can compute $\alpha$ for a given $\sigma^2$ and $\beta$ like so:
+So if you do the algebra, you'll see that you can compute $\alpha$ for a given $\sigma^2$ and $\beta$ with the equation
 
-$$\alpha = \sqrt{ \frac{\sigma^2 \Gamma (1/\beta)}{\Gamma (3/\beta)} }$$
+$$\alpha = \sqrt{ \frac{\sigma^2 \Gamma (1/\beta)}{\Gamma (3/\beta)} }.$$
 
 I got the formula from [Wikipedia.com](https://en.wikipedia.org/wiki/Generalized_normal_distribution). Don't judge. We can wrap that formula in a custom function, `alpha_per_beta()`, use it to solve for the desired $\beta$ values, and plot. But one more thing: McElreath didn't tell us exactly which $\beta$ values the left panel of Figure 9.2 was based on. So the plot below is my best guess.