From a80c4a173bfe1acfbe51018f5d1d36e06f4865b7 Mon Sep 17 00:00:00 2001
From: mmcky <mamckay@gmail.com>
Date: Fri, 26 Apr 2024 15:09:45 +1000
Subject: [PATCH 01/10] hide yf.download output

---
 lectures/heavy_tails.md | 9 +++++++--
 1 file changed, 7 insertions(+), 2 deletions(-)

diff --git a/lectures/heavy_tails.md b/lectures/heavy_tails.md
index 639e6b1d..a0680009 100644
--- a/lectures/heavy_tails.md
+++ b/lectures/heavy_tails.md
@@ -4,7 +4,7 @@ jupytext:
     extension: .md
     format_name: myst
     format_version: 0.13
-    jupytext_version: 1.14.5
+    jupytext_version: 1.16.1
 kernelspec:
   display_name: Python 3 (ipykernel)
   language: python
@@ -169,7 +169,12 @@ This equates to daily returns if we set dividends aside.
 The code below produces the desired plot using Yahoo financial data via the `yfinance` library.
 
 ```{code-cell} ipython3
-s = yf.download('AMZN', '2015-1-1', '2022-7-1')['Adj Close']
+:tags: [hide-output]
+data = yf.download('AMZN', '2015-1-1', '2022-7-1')
+```
+
+```{code-cell} ipython3
+s = data['Adj Close']
 r = s.pct_change()
 
 fig, ax = plt.subplots()

From 536a511292c0e26d5d493fc3741f67b4c8b21929 Mon Sep 17 00:00:00 2001
From: mmcky <mamckay@gmail.com>
Date: Fri, 26 Apr 2024 15:31:30 +1000
Subject: [PATCH 02/10] reformat display math, add mystnb figure tags

---
 lectures/heavy_tails.md | 145 +++++++++++++++++++++++++++++++++-------
 1 file changed, 121 insertions(+), 24 deletions(-)

diff --git a/lectures/heavy_tails.md b/lectures/heavy_tails.md
index a0680009..533c8cbf 100644
--- a/lectures/heavy_tails.md
+++ b/lectures/heavy_tails.md
@@ -61,10 +61,10 @@ To explain this concept, let's look first at examples.
 The classic example is the [normal distribution](https://en.wikipedia.org/wiki/Normal_distribution), which has density
 
 $$ 
-    f(x) = \frac{1}{\sqrt{2\pi}\sigma} 
-    \exp\left( -\frac{(x-\mu)^2}{2 \sigma^2} \right)
-    \qquad
-    (-\infty < x < \infty)
+f(x) = \frac{1}{\sqrt{2\pi}\sigma} 
+\exp\left( -\frac{(x-\mu)^2}{2 \sigma^2} \right)
+\qquad
+(-\infty < x < \infty)
 $$
 
 
@@ -78,6 +78,12 @@ We can see this when we plot the density and show a histogram of observations,
 as with the following code (which assumes $\mu=0$ and $\sigma=1$).
 
 ```{code-cell} ipython3
+---
+mystnb:
+  figure:
+    caption: Histogram of observations
+    name: hist-obs
+---
 fig, ax = plt.subplots()
 X = norm.rvs(size=1_000_000)
 ax.hist(X, bins=40, alpha=0.4, label='histogram', density=True)
@@ -101,6 +107,12 @@ X.min(), X.max()
 Here's another view of draws from the same distribution:
 
 ```{code-cell} ipython3
+---
+mystnb:
+  figure:
+    caption: Histogram of observations
+    name: hist-obs2
+---
 n = 2000
 fig, ax = plt.subplots()
 data = norm.rvs(size=n)
@@ -174,6 +186,12 @@ data = yf.download('AMZN', '2015-1-1', '2022-7-1')
 ```
 
 ```{code-cell} ipython3
+---
+mystnb:
+  figure:
+    caption: Daily Amazon returns
+    name: dailyreturns-amzn
+---
 s = data['Adj Close']
 r = s.pct_change()
 
@@ -194,7 +212,18 @@ Several of observations are quite extreme.
 We get a similar picture if we look at other assets, such as Bitcoin
 
 ```{code-cell} ipython3
-s = yf.download('BTC-USD', '2015-1-1', '2022-7-1')['Adj Close']
+:tags: [hide-output]
+data = yf.download('BTC-USD', '2015-1-1', '2022-7-1')
+```
+
+```{code-cell} ipython3
+---
+mystnb:
+  figure:
+    caption: Daily Bitcoin returns
+    name: dailyreturns-btc
+---
+s = data['Adj Close']
 r = s.pct_change()
 
 fig, ax = plt.subplots()
@@ -211,6 +240,12 @@ The histogram also looks different to the histogram of the normal
 distribution:
 
 ```{code-cell} ipython3
+---
+mystnb:
+  figure:
+    caption: Histogram (Normal vs Bitcoin returns)
+    name: hist-normal-btc
+---
 r = np.random.standard_t(df=5, size=1000)
 
 fig, ax = plt.subplots()
@@ -274,10 +309,6 @@ like
 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.
 
@@ -300,6 +331,12 @@ distribution](https://en.wikipedia.org/wiki/Cauchy_distribution), which is
 heavy-tailed.
 
 ```{code-cell} ipython3
+---
+mystnb:
+  figure:
+    caption: Histogram of Cauchy distribution
+    name: hist-cauchy
+---
 n = 120
 np.random.seed(11)
 
@@ -353,6 +390,12 @@ The exponential distribution is a light-tailed distribution.
 Here are some draws from the exponential distribution.
 
 ```{code-cell} ipython3
+---
+mystnb:
+  figure:
+    caption: Histogram of Exponential distribution
+    name: hist-exponential
+---
 n = 120
 np.random.seed(11)
 
@@ -394,7 +437,9 @@ exponential random variable.
 
 In particular, if $X$ is exponentially distributed with rate parameter $\alpha$, then
 
-$$   Y = \bar x \exp(X) $$
+$$
+Y = \bar x \exp(X) 
+$$
 
 is Pareto-distributed with minimum $\bar x$ and tail index $\alpha$. 
 
@@ -402,6 +447,12 @@ Here are some draws from the Pareto distribution with tail index $1$ and minimum
 $1$.
 
 ```{code-cell} ipython3
+---
+mystnb:
+  figure:
+    caption: Histogram of Pareto distribution
+    name: hist-pareto
+---
 n = 120
 np.random.seed(11)
 
@@ -425,7 +476,9 @@ light and heavy tails is to look at the
 
 For a random variable $X$ with CDF $F$, the CCDF is the function 
 
-$$ G(x) := 1 - F(x) = \mathbb P\{X > x\} $$
+$$
+G(x) := 1 - F(x) = \mathbb P\{X > x\} 
+$$
 
 (Some authors call $G$ the "survival" function.)
 
@@ -433,13 +486,17 @@ The CCDF shows how fast the upper tail goes to zero as $x \to \infty$.
 
 If $X$ is exponentially distributed with rate parameter $\alpha$, then the CCDF is
 
-$$ G_E(x) = \exp(- \alpha x) $$
+$$
+G_E(x) = \exp(- \alpha x)
+$$
 
 This function goes to zero relatively quickly as $x$ gets large.
 
 The standard Pareto distribution, where $\bar x = 1$, has CCDF
 
-$$ G_P(x) = x^{- \alpha} $$
+$$
+G_P(x) = x^{- \alpha}
+$$
 
 This function goes to zero as $x \to \infty$, but much slower than $G_E$.
 
@@ -505,13 +562,21 @@ The sample counterpart of the CCDF function is the **empirical CCDF**.
 
 Given a sample $x_1, \ldots, x_n$, the empirical CCDF is given by
 
-$$ \hat G(x) = \frac{1}{n} \sum_{i=1}^n \mathbb 1\{x_i > x\} $$
+$$
+\hat G(x) = \frac{1}{n} \sum_{i=1}^n \mathbb 1\{x_i > x\}
+$$
 
 Thus, $\hat G(x)$ shows the fraction of the sample that exceeds $x$.
 
 Here's a figure containing some empirical CCDFs from simulated data.
 
 ```{code-cell} ipython3
+---
+mystnb:
+  figure:
+    caption: Empirical CCDFs
+    name: ccdf-empirics
+---
 def eccdf(x, data):
     "Simple empirical CCDF function."
     return np.mean(data > x)
@@ -690,7 +755,13 @@ def extract_wb(varlist=['NY.GDP.MKTP.CD'],
 Here is a plot of the firm size distribution for the largest 500 firms in 2020 taken from Forbes Global 2000.
 
 ```{code-cell} ipython3
-:tags: [hide-input]
+---
+tags: [hide-input]
+mystnb:
+  figure:
+    caption: Firm size distribution
+    name: firm-size-dist
+--- 
 
 df_fs = pd.read_csv('https://media.githubusercontent.com/media/QuantEcon/high_dim_data/main/cross_section/forbes-global2000.csv')
 df_fs = df_fs[['Country', 'Sales', 'Profits', 'Assets', 'Market Value']]
@@ -711,7 +782,13 @@ Here are plots of the city size distribution for the US and Brazil in 2023 from
 The size is measured by population.
 
 ```{code-cell} ipython3
-:tags: [hide-input]
+---
+tags: [hide-input]
+mystnb:
+  figure:
+    caption: City size distribution
+    name: city-size-dist
+--- 
 
 # import population data of cities in 2023 United States and 2023 Brazil from world population review
 df_cs_us = pd.read_csv('https://media.githubusercontent.com/media/QuantEcon/high_dim_data/main/cross_section/cities_us.csv')
@@ -732,7 +809,13 @@ Here is a plot of the upper tail (top 500) of the wealth distribution.
 The data is from the Forbes Billionaires list in 2020.
 
 ```{code-cell} ipython3
-:tags: [hide-input]
+---
+tags: [hide-input]
+mystnb:
+  figure:
+    caption: Wealth distribution (Forbes Billionaires in 2020)
+    name: wealth-dist
+--- 
 
 df_w = pd.read_csv('https://media.githubusercontent.com/media/QuantEcon/high_dim_data/main/cross_section/forbes-billionaires.csv')
 df_w = df_w[['country', 'realTimeWorth', 'realTimeRank']].dropna()
@@ -782,7 +865,13 @@ df_gdp1.dropna(inplace=True)
 ```
 
 ```{code-cell} ipython3
-:tags: [hide-input]
+---
+tags: [hide-input]
+mystnb:
+  figure:
+    caption: GDP per capita distribution
+    name: gdppc-dist
+--- 
 
 fig, axes = plt.subplots(1, 2, figsize=(8.8, 3.6))
 
@@ -828,6 +917,12 @@ Let's have a look at the behavior of the sample mean in this case, and see
 whether or not the LLN is still valid.
 
 ```{code-cell} ipython3
+---
+mystnb:
+  figure:
+    caption: LLN failure
+    name: fail-lln
+--- 
 from scipy.stats import cauchy
 
 np.random.seed(1234)
@@ -887,7 +982,9 @@ portfolio is $\mu$ and the variance is $\sigma^2$.
 
 If instead the investor puts share $1/n$ of her wealth in each asset, then the portfolio payoff is
 
-$$ Y_n = \sum_{i=1}^n \frac{X_i}{n} = \frac{1}{n} \sum_{i=1}^n X_i. $$  
+$$
+Y_n = \sum_{i=1}^n \frac{X_i}{n} = \frac{1}{n} \sum_{i=1}^n X_i. 
+$$  
 
 Try computing the mean and variance.
 
@@ -918,8 +1015,6 @@ For example, the heaviness of the tail of the income distribution helps
 determine {doc}`how much revenue a given tax policy will raise <mle>`.
 
 
-
-
 (cltail)=
 ## Classifying tail properties
 
@@ -964,7 +1059,9 @@ For example, every random variable with bounded support is light-tailed. (Why?)
 
 As another example, if $X$ has the [exponential distribution](https://en.wikipedia.org/wiki/Exponential_distribution), with cdf $F(x) = 1 - \exp(-\lambda x)$ for some $\lambda > 0$, then its moment generating function is 
 
-$$ m(t) = \frac{\lambda}{\lambda - t} \quad \text{when } t < \lambda $$
+$$
+m(t) = \frac{\lambda}{\lambda - t} \quad \text{when } t < \lambda 
+$$
 
 In particular, $m(t)$ is finite whenever $t < \lambda$, so $X$ is light-tailed.
 
@@ -1023,7 +1120,7 @@ $$
 But then
 
 $$
-    \mathbb E X^r = r \int_0^\infty x^{r-1} \mathbb P\{ X > x \} dx
+\mathbb E X^r = r \int_0^\infty x^{r-1} \mathbb P\{ X > x \} dx
 \geq
 r \int_0^{\bar x} x^{r-1} \mathbb P\{ X > x \} dx
 + r \int_{\bar x}^\infty  x^{r-1} b x^{-\alpha} dx.
@@ -1254,7 +1351,7 @@ assumption leads to a lower mean and greater dispersion.
 The [characteristic function](https://en.wikipedia.org/wiki/Characteristic_function_%28probability_theory%29) of the Cauchy distribution is
 
 $$
-    \phi(t) = \mathbb E e^{itX} = \int e^{i t x} f(x) dx = e^{-|t|}
+\phi(t) = \mathbb E e^{itX} = \int e^{i t x} f(x) dx = e^{-|t|}
 $$ (lln_cch)
 
 Prove that the sample mean $\bar X_n$ of $n$ independent draws $X_1, \ldots,

From b135dd83271d59f8bc136e1feb537b8feb2cc48e Mon Sep 17 00:00:00 2001
From: mmcky <mamckay@gmail.com>
Date: Fri, 26 Apr 2024 15:50:12 +1000
Subject: [PATCH 03/10] review of figure titles

---
 lectures/heavy_tails.md | 24 ++++++++++++++++++------
 1 file changed, 18 insertions(+), 6 deletions(-)

diff --git a/lectures/heavy_tails.md b/lectures/heavy_tails.md
index 533c8cbf..42042630 100644
--- a/lectures/heavy_tails.md
+++ b/lectures/heavy_tails.md
@@ -334,8 +334,8 @@ heavy-tailed.
 ---
 mystnb:
   figure:
-    caption: Histogram of Cauchy distribution
-    name: hist-cauchy
+    caption: Draws from Normal and Cauchy distributions
+    name: draws-normal-cauchy
 ---
 n = 120
 np.random.seed(11)
@@ -393,8 +393,8 @@ Here are some draws from the exponential distribution.
 ---
 mystnb:
   figure:
-    caption: Histogram of Exponential distribution
-    name: hist-exponential
+    caption: Draws of Exponential distribution
+    name: draws-exponential
 ---
 n = 120
 np.random.seed(11)
@@ -450,8 +450,8 @@ $1$.
 ---
 mystnb:
   figure:
-    caption: Histogram of Pareto distribution
-    name: hist-pareto
+    caption: Draws from Pareto distribution
+    name: draws-pareto
 ---
 n = 120
 np.random.seed(11)
@@ -528,6 +528,12 @@ $$
 Here's a plot that illustrates how $G_E$ goes to zero faster than $G_P$.
 
 ```{code-cell} ipython3
+---
+mystnb:
+  figure:
+    caption: Pareto and exponential distribution comparison
+    name: compare-pareto-exponential
+---
 x = np.linspace(1.5, 100, 1000)
 fig, ax = plt.subplots()
 alpha = 1.0
@@ -541,6 +547,12 @@ Here's a log-log plot of the same functions, which makes visual comparison
 easier.
 
 ```{code-cell} ipython3
+---
+mystnb:
+  figure:
+    caption: Pareto and exponential distribution comparison (log-log)
+    name: compare-pareto-exponential-log-log
+---
 fig, ax = plt.subplots()
 alpha = 1.0
 ax.loglog(x, np.exp(- alpha * x), label='exponential', alpha=0.8)

From b607dd115a33cb5b0db86b7ef39dfea7f0c3b272 Mon Sep 17 00:00:00 2001
From: mmcky <mamckay@gmail.com>
Date: Mon, 29 Apr 2024 10:37:44 +1000
Subject: [PATCH 04/10] adjust line colour

---
 lectures/heavy_tails.md | 22 ++++++++++++----------
 1 file changed, 12 insertions(+), 10 deletions(-)

diff --git a/lectures/heavy_tails.md b/lectures/heavy_tails.md
index 42042630..29008d6f 100644
--- a/lectures/heavy_tails.md
+++ b/lectures/heavy_tails.md
@@ -182,6 +182,7 @@ The code below produces the desired plot using Yahoo financial data via the `yfi
 
 ```{code-cell} ipython3
 :tags: [hide-output]
+
 data = yf.download('AMZN', '2015-1-1', '2022-7-1')
 ```
 
@@ -213,6 +214,7 @@ We get a similar picture if we look at other assets, such as Bitcoin
 
 ```{code-cell} ipython3
 :tags: [hide-output]
+
 data = yf.download('BTC-USD', '2015-1-1', '2022-7-1')
 ```
 
@@ -254,7 +256,7 @@ 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.plot(x, p, linewidth=2, label='normal distribution')
 
 ax.set_xlabel('returns', fontsize=12)
 ax.legend()
@@ -768,12 +770,12 @@ Here is a plot of the firm size distribution for the largest 500 firms in 2020 t
 
 ```{code-cell} ipython3
 ---
-tags: [hide-input]
 mystnb:
   figure:
     caption: Firm size distribution
     name: firm-size-dist
---- 
+tags: [hide-input]
+---
 
 df_fs = pd.read_csv('https://media.githubusercontent.com/media/QuantEcon/high_dim_data/main/cross_section/forbes-global2000.csv')
 df_fs = df_fs[['Country', 'Sales', 'Profits', 'Assets', 'Market Value']]
@@ -795,12 +797,12 @@ The size is measured by population.
 
 ```{code-cell} ipython3
 ---
-tags: [hide-input]
 mystnb:
   figure:
     caption: City size distribution
     name: city-size-dist
---- 
+tags: [hide-input]
+---
 
 # import population data of cities in 2023 United States and 2023 Brazil from world population review
 df_cs_us = pd.read_csv('https://media.githubusercontent.com/media/QuantEcon/high_dim_data/main/cross_section/cities_us.csv')
@@ -822,12 +824,12 @@ The data is from the Forbes Billionaires list in 2020.
 
 ```{code-cell} ipython3
 ---
-tags: [hide-input]
 mystnb:
   figure:
     caption: Wealth distribution (Forbes Billionaires in 2020)
     name: wealth-dist
---- 
+tags: [hide-input]
+---
 
 df_w = pd.read_csv('https://media.githubusercontent.com/media/QuantEcon/high_dim_data/main/cross_section/forbes-billionaires.csv')
 df_w = df_w[['country', 'realTimeWorth', 'realTimeRank']].dropna()
@@ -878,12 +880,12 @@ df_gdp1.dropna(inplace=True)
 
 ```{code-cell} ipython3
 ---
-tags: [hide-input]
 mystnb:
   figure:
     caption: GDP per capita distribution
     name: gdppc-dist
---- 
+tags: [hide-input]
+---
 
 fig, axes = plt.subplots(1, 2, figsize=(8.8, 3.6))
 
@@ -934,7 +936,7 @@ mystnb:
   figure:
     caption: LLN failure
     name: fail-lln
---- 
+---
 from scipy.stats import cauchy
 
 np.random.seed(1234)

From 959e37da26bbb83e9c439285f38e4db9e5acb280 Mon Sep 17 00:00:00 2001
From: mmcky <mamckay@gmail.com>
Date: Mon, 29 Apr 2024 11:10:22 +1000
Subject: [PATCH 05/10] reorganise code and add q-q plot

---
 lectures/heavy_tails.md | 55 ++++++++++++++++++++++++++++++++---------
 1 file changed, 43 insertions(+), 12 deletions(-)

diff --git a/lectures/heavy_tails.md b/lectures/heavy_tails.md
index 29008d6f..b3a90022 100644
--- a/lectures/heavy_tails.md
+++ b/lectures/heavy_tails.md
@@ -582,6 +582,12 @@ $$
 
 Thus, $\hat G(x)$ shows the fraction of the sample that exceeds $x$.
 
+```{code-cell} ipython3
+def eccdf(x, data):
+    "Simple empirical CCDF function."
+    return np.mean(data > x)
+```
+
 Here's a figure containing some empirical CCDFs from simulated data.
 
 ```{code-cell} ipython3
@@ -591,21 +597,20 @@ mystnb:
     caption: Empirical CCDFs
     name: ccdf-empirics
 ---
-def eccdf(x, data):
-    "Simple empirical CCDF function."
-    return np.mean(data > x)
-
+# Parameters and grid
 x_grid = np.linspace(1, 1000, 1000)
 sample_size = 1000
 np.random.seed(13)
 z = np.random.randn(sample_size)
 
-data_1 = np.random.exponential(size=sample_size)
-data_2 = np.exp(z)
-data_3 = np.exp(np.random.exponential(size=sample_size))
+# Draws
+data_exp = np.random.exponential(size=sample_size)
+data_logn = np.exp(z)
+data_pareto = np.exp(np.random.exponential(size=sample_size))
 
-data_list = [data_1, data_2, data_3]
+data_list = [data_exp, data_logn, data_pareto]
 
+# Build figure
 fig, axes = plt.subplots(3, 1, figsize=(6, 8))
 axes = axes.flatten()
 labels = ['exponential', 'lognormal', 'Pareto']
@@ -630,6 +635,36 @@ approximately linear in a log-log plot.
 
 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.
 
++++
+
+#### Q-Q Plots
+
+We can also use a [qq plot](https://en.wikipedia.org/wiki/Q%E2%80%93Q_plot) to do a visual comparison between two probability distributions. 
+
+The [statsmodels](https://www.statsmodels.org/stable/index.html) package provides a convenient [qqplot](https://www.statsmodels.org/stable/generated/statsmodels.graphics.gofplots.qqplot.html) function that, by default, compares sample data to the quintiles of the normal distribution.
+
+If the data is drawn from a Normal distribution, the plot would look like:
+
+```{code-cell} ipython3
+data_normal = np.random.normal(size=sample_size)
+sm.qqplot(data_normal, line='45')
+plt.show()
+```
+
+We can now compare this with the exponential, log-normal, and pareto distributions
+
+```{code-cell} ipython3
+# Build figure
+fig, axes = plt.subplots(3, 1, figsize=(6, 8))
+axes = axes.flatten()
+labels = ['exponential', 'lognormal', 'Pareto']
+for data, label, ax in zip(data_list, labels, axes):
+    sm.qqplot(data, line='45', ax=ax, )
+    ax.set_title(label)
+plt.tight_layout()
+plt.show()
+```
+
 
 ### Power laws 
 
@@ -776,7 +811,6 @@ mystnb:
     name: firm-size-dist
 tags: [hide-input]
 ---
-
 df_fs = pd.read_csv('https://media.githubusercontent.com/media/QuantEcon/high_dim_data/main/cross_section/forbes-global2000.csv')
 df_fs = df_fs[['Country', 'Sales', 'Profits', 'Assets', 'Market Value']]
 fig, ax = plt.subplots(figsize=(6.4, 3.5))
@@ -803,7 +837,6 @@ mystnb:
     name: city-size-dist
 tags: [hide-input]
 ---
-
 # import population data of cities in 2023 United States and 2023 Brazil from world population review
 df_cs_us = pd.read_csv('https://media.githubusercontent.com/media/QuantEcon/high_dim_data/main/cross_section/cities_us.csv')
 df_cs_br = pd.read_csv('https://media.githubusercontent.com/media/QuantEcon/high_dim_data/main/cross_section/cities_brazil.csv')
@@ -830,7 +863,6 @@ mystnb:
     name: wealth-dist
 tags: [hide-input]
 ---
-
 df_w = pd.read_csv('https://media.githubusercontent.com/media/QuantEcon/high_dim_data/main/cross_section/forbes-billionaires.csv')
 df_w = df_w[['country', 'realTimeWorth', 'realTimeRank']].dropna()
 df_w = df_w.astype({'realTimeRank': int})
@@ -886,7 +918,6 @@ mystnb:
     name: gdppc-dist
 tags: [hide-input]
 ---
-
 fig, axes = plt.subplots(1, 2, figsize=(8.8, 3.6))
 
 for name, ax in zip(variable_names, axes):

From d6f575003fbc5cada05690a3f5e93ccfb1687ff5 Mon Sep 17 00:00:00 2001
From: mmcky <mamckay@gmail.com>
Date: Mon, 29 Apr 2024 11:28:03 +1000
Subject: [PATCH 06/10] add wiki links and idea on less formal discussion on
 heavy-tails

---
 lectures/heavy_tails.md | 4 +++-
 1 file changed, 3 insertions(+), 1 deletion(-)

diff --git a/lectures/heavy_tails.md b/lectures/heavy_tails.md
index b3a90022..80c4f75a 100644
--- a/lectures/heavy_tails.md
+++ b/lectures/heavy_tails.md
@@ -1076,7 +1076,7 @@ left hand tails are very similar and we omit them to simplify the exposition.
 
 ### Light and heavy tails
 
-A distribution $F$ with density $f$ on $\mathbb R_+$ is called **heavy-tailed** if
+A distribution $F$ with density $f$ on $\mathbb R_+$ is called [heavy-tailed](https://en.wikipedia.org/wiki/Heavy-tailed_distribution) if
 
 ```{math}
 :label: defht
@@ -1096,6 +1096,8 @@ $(0, \infty)$.
 
 The Pareto distribution is also heavy-tailed.
 
+Less formally, a **heavy-tailed** distribution is one that is not exponentially bounded (i.e. the tails are heavier than the exponential distribution). 
+
 A distribution $F$ on $\mathbb R_+$ is called **light-tailed** if it is not heavy-tailed.
 
 A nonnegative random variable $X$ is **light-tailed** if its distribution $F$ is light-tailed.

From 15f9e14d293d21ab094b28364800fb232a1ebf06 Mon Sep 17 00:00:00 2001
From: mmcky <mamckay@gmail.com>
Date: Mon, 29 Apr 2024 11:33:50 +1000
Subject: [PATCH 07/10] square q-q plots

---
 lectures/heavy_tails.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/lectures/heavy_tails.md b/lectures/heavy_tails.md
index 80c4f75a..b01ebbea 100644
--- a/lectures/heavy_tails.md
+++ b/lectures/heavy_tails.md
@@ -655,7 +655,7 @@ We can now compare this with the exponential, log-normal, and pareto distributio
 
 ```{code-cell} ipython3
 # Build figure
-fig, axes = plt.subplots(3, 1, figsize=(6, 8))
+fig, axes = plt.subplots(3, 1, figsize=(8, 8))
 axes = axes.flatten()
 labels = ['exponential', 'lognormal', 'Pareto']
 for data, label, ax in zip(data_list, labels, axes):

From 90bf2c26af2f1fbe2524d1528702bdec6ae79b69 Mon Sep 17 00:00:00 2001
From: mmcky <mamckay@gmail.com>
Date: Mon, 29 Apr 2024 11:56:12 +1000
Subject: [PATCH 08/10] qq-plot multiples of 3 and horizontal

---
 lectures/heavy_tails.md | 3 +--
 1 file changed, 1 insertion(+), 2 deletions(-)

diff --git a/lectures/heavy_tails.md b/lectures/heavy_tails.md
index b01ebbea..a63bd34b 100644
--- a/lectures/heavy_tails.md
+++ b/lectures/heavy_tails.md
@@ -655,7 +655,7 @@ We can now compare this with the exponential, log-normal, and pareto distributio
 
 ```{code-cell} ipython3
 # Build figure
-fig, axes = plt.subplots(3, 1, figsize=(8, 8))
+fig, axes = plt.subplots(1, 3, figsize=(12, 4))
 axes = axes.flatten()
 labels = ['exponential', 'lognormal', 'Pareto']
 for data, label, ax in zip(data_list, labels, axes):
@@ -665,7 +665,6 @@ plt.tight_layout()
 plt.show()
 ```
 
-
 ### Power laws 
 
 

From f32785b73f4e3377fb6557bd5ebcddc4b662c79a Mon Sep 17 00:00:00 2001
From: mmcky <mamckay@gmail.com>
Date: Mon, 29 Apr 2024 14:28:13 +1000
Subject: [PATCH 09/10] follow wikipedia style for capitalization, Normal ->
 normal in text

---
 lectures/heavy_tails.md | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

diff --git a/lectures/heavy_tails.md b/lectures/heavy_tails.md
index a63bd34b..0dd48009 100644
--- a/lectures/heavy_tails.md
+++ b/lectures/heavy_tails.md
@@ -245,7 +245,7 @@ distribution:
 ---
 mystnb:
   figure:
-    caption: Histogram (Normal vs Bitcoin returns)
+    caption: Histogram (normal vs bitcoin returns)
     name: hist-normal-btc
 ---
 r = np.random.standard_t(df=5, size=1000)
@@ -336,7 +336,7 @@ heavy-tailed.
 ---
 mystnb:
   figure:
-    caption: Draws from Normal and Cauchy distributions
+    caption: Draws from normal and Cauchy distributions
     name: draws-normal-cauchy
 ---
 n = 120
@@ -395,7 +395,7 @@ Here are some draws from the exponential distribution.
 ---
 mystnb:
   figure:
-    caption: Draws of Exponential distribution
+    caption: Draws of exponential distribution
     name: draws-exponential
 ---
 n = 120
@@ -643,7 +643,7 @@ We can also use a [qq plot](https://en.wikipedia.org/wiki/Q%E2%80%93Q_plot) to d
 
 The [statsmodels](https://www.statsmodels.org/stable/index.html) package provides a convenient [qqplot](https://www.statsmodels.org/stable/generated/statsmodels.graphics.gofplots.qqplot.html) function that, by default, compares sample data to the quintiles of the normal distribution.
 
-If the data is drawn from a Normal distribution, the plot would look like:
+If the data is drawn from a normal distribution, the plot would look like:
 
 ```{code-cell} ipython3
 data_normal = np.random.normal(size=sample_size)
@@ -651,7 +651,7 @@ sm.qqplot(data_normal, line='45')
 plt.show()
 ```
 
-We can now compare this with the exponential, log-normal, and pareto distributions
+We can now compare this with the exponential, log-normal, and Pareto distributions
 
 ```{code-cell} ipython3
 # Build figure
@@ -858,7 +858,7 @@ The data is from the Forbes Billionaires list in 2020.
 ---
 mystnb:
   figure:
-    caption: Wealth distribution (Forbes Billionaires in 2020)
+    caption: Wealth distribution (Forbes billionaires in 2020)
     name: wealth-dist
 tags: [hide-input]
 ---

From a44a5df4b131965fb7e83f184b8adf9ec53b79fa Mon Sep 17 00:00:00 2001
From: mmcky <mamckay@gmail.com>
Date: Tue, 30 Apr 2024 10:34:43 +1000
Subject: [PATCH 10/10] remove bolding as not formal

---
 lectures/heavy_tails.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/lectures/heavy_tails.md b/lectures/heavy_tails.md
index 0dd48009..137af816 100644
--- a/lectures/heavy_tails.md
+++ b/lectures/heavy_tails.md
@@ -1095,7 +1095,7 @@ $(0, \infty)$.
 
 The Pareto distribution is also heavy-tailed.
 
-Less formally, a **heavy-tailed** distribution is one that is not exponentially bounded (i.e. the tails are heavier than the exponential distribution). 
+Less formally, a heavy-tailed distribution is one that is not exponentially bounded (i.e. the tails are heavier than the exponential distribution). 
 
 A distribution $F$ on $\mathbb R_+$ is called **light-tailed** if it is not heavy-tailed.