[heavy_tail] Update editorial suggestions

lectures/heavy_tails.md

@@ -39,6 +39,12 @@ register_matplotlib_converters()
 
 ## Overview
 
+Studying heavy-tailed distributions is essential for accurately comprehending real-world phenomena. 
+
+Unlike standard Gaussian distributions, heavy-tailed distributions account for extreme events with greater probabilities. 
+
+This understanding is crucial in analyzing wealth, firm size, and city size distributions, as well as other areas such as business cycles and political economy.
+
 In this section we give some motivation for the lecture.
 
 ### Introduction: light tails
@@ -112,7 +118,7 @@ too much from the mean.
 Putting this another way, light-tailed distributions are those that
 rarely generate extreme values.
 
-(A more formal definition is given below.)
+(A more formal definition is given [below](https://intro.quantecon.org/heavy_tails.html#light-and-heavy-tails).)
 
 Many statisticians and econometricians 
 use rules of thumb such as "outcomes more than four or five
@@ -196,9 +202,19 @@ The histogram also looks different to the histogram of the normal
 distribution:
 
 ```{code-cell} ipython3
+r = np.random.standard_t(df=5, size=1000)
+
 fig, ax = plt.subplots()
 ax.hist(r, bins=60, alpha=0.4, label='bitcoin returns', density=True)
+
+xmin, xmax = plt.xlim()
+x = np.linspace(xmin, xmax, 100)
+p = norm.pdf(x, np.mean(r), np.std(r))
+ax.plot(x, p, 'k', linewidth=2, label='normal distribution')
+
 ax.set_xlabel('returns', fontsize=12)
+ax.legend()
+
 plt.show()
 ```
 
@@ -246,14 +262,15 @@ like
 * forecasting
 * taxation (across a heavy-tailed income distribution), etc.
 
-We return to these points below.
+We return to these points [below](https://intro.quantecon.org/heavy_tails.html#why-do-heavy-tails-matter).
 
 
 
 
 
 
 ## Visual comparisons
+In this section, we will introduce important concepts such as the Pareto distribution, Counter CDFs, and Power laws, which aid in recognizing heavy-tailed distributions.
 
 Later we will provide a mathematical definition of the difference between
 light and heavy tails.
@@ -523,7 +540,7 @@ plt.show()
 As with the CCDF, the empirical CCDF from the Pareto distributions is 
 approximately linear in a log-log plot.
 
-We will use this idea below when we look at real data.
+We will use this idea [below](https://intro.quantecon.org/heavy_tails.html#heavy-tails-in-economic-cross-sections) when we look at real data.
 
 
 ### Power laws