@@ -118,7 +118,7 @@ too much from the mean.
118118Putting this another way, light-tailed distributions are those that
119119rarely generate extreme values.
120120
121- (A more formal definition is given [ below] ( https://intro.quantecon.org/heavy_tails.html#light-and- heavy-tails ) .)
121+ (A more formal definition is given {ref} ` below < heavy-tail:formal-definition> ` .)
122122
123123Many statisticians and econometricians
124124use rules of thumb such as "outcomes more than four or five
262262* forecasting
263263* taxation (across a heavy-tailed income distribution), etc.
264264
265- We return to these points [ below] ( https://intro.quantecon.org/heavy_tails.html#why-do- heavy-tails-matter ) .
265+ We return to these points {ref} ` below < heavy-tail:application> ` .
266266
267267
268268
@@ -467,6 +467,8 @@ fig, ax = plt.subplots()
467467alpha = 1.0
468468ax.plot(x, np.exp(- alpha * x), label='exponential', alpha=0.8)
469469ax.plot(x, x**(- alpha), label='Pareto', alpha=0.8)
470+ ax.set_xlabel('X value', fontsize=12)
471+ ax.set_ylabel('CCDF', fontsize=12)
470472ax.legend()
471473plt.show()
472474```
@@ -479,6 +481,8 @@ fig, ax = plt.subplots()
479481alpha = 1.0
480482ax.loglog(x, np.exp(- alpha * x), label='exponential', alpha=0.8)
481483ax.loglog(x, x**(- alpha), label='Pareto', alpha=0.8)
484+ ax.set_xlabel('log value', fontsize=12)
485+ ax.set_ylabel('log prob', fontsize=12)
482486ax.legend()
483487plt.show()
484488```
@@ -848,6 +852,7 @@ The sequence shows no sign of converging.
848852We return to this point in the exercises.
849853
850854
855+ (heavy-tail: application )=
851856## Why do heavy tails matter?
852857
853858We have now seen that
@@ -925,6 +930,7 @@ nonnegative random variables and their distributions.
925930The definitions for
926931left hand tails are very similar and we omit them to simplify the exposition.
927932
933+ (heavy-tail: formal-definition )=
928934### Light and heavy tails
929935
930936A distribution $F$ with density $f$ on $\mathbb R_ +$ is called ** heavy-tailed** if
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