typos
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@@ -228,6 +228,7 @@ Note that the function <b>multivariate</b> returns also the covariance discussed
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">IPython.display</span> <span style="color: #008000; font-weight: bold">import</span> display
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n <span style="color: #666666">=</span> <span style="color: #666666">10000</span>
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mean <span style="color: #666666">=</span> (<span style="color: #666666">-1</span>, <span style="color: #666666">2</span>)
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@@ -298,8 +299,8 @@ Our own code here is not very elegant and asks for improvements.
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># extract the relevant columns from the centered design matrix of dim n x 2 x = X_centered[:,[0]]</span>
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y <span style="color: #666666">=</span> X_centered[:,[<span style="color: #666666">1</span>]]
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># extract the relevant columns from the centered design matrix of dim n x 2 x = X_centered[:,0]</span>
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y <span style="color: #666666">=</span> X_centered[:,<span style="color: #666666">1</span>]
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Cov <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #666666">2</span>,<span style="color: #666666">2</span>))
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Cov[<span style="color: #666666">0</span>,<span style="color: #666666">1</span>] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(x<span style="color: #666666">.</span>T<span style="color: #AA22FF">@y</span>)<span style="color: #666666">/</span>(n<span style="color: #666666">-1.0</span>)
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Cov[<span style="color: #666666">0</span>,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(x<span style="color: #666666">.</span>T<span style="color: #AA22FF">@x</span>)<span style="color: #666666">/</span>(n<span style="color: #666666">-1.0</span>)
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@@ -1034,6 +1034,7 @@ Note that the function <b>multivariate</b> returns also the covariance discussed
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<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
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<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
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<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">pandas</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">pd</span>
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<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">IPython.display</span> <span style="color: #8B008B; font-weight: bold">import</span> display
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n = <span style="color: #B452CD">10000</span>
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mean = (-<span style="color: #B452CD">1</span>, <span style="color: #B452CD">2</span>)
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@@ -1110,8 +1111,8 @@ Our own code here is not very elegant and asks for improvements.
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
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<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #228B22"># extract the relevant columns from the centered design matrix of dim n x 2 x = X_centered[:,[0]]</span>
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y = X_centered[:,[<span style="color: #B452CD">1</span>]]
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<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #228B22"># extract the relevant columns from the centered design matrix of dim n x 2 x = X_centered[:,0]</span>
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y = X_centered[:,<span style="color: #B452CD">1</span>]
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Cov = np.zeros((<span style="color: #B452CD">2</span>,<span style="color: #B452CD">2</span>))
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Cov[<span style="color: #B452CD">0</span>,<span style="color: #B452CD">1</span>] = np.sum(x.T<span style="color: #707a7c">@y</span>)/(n-<span style="color: #B452CD">1.0</span>)
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Cov[<span style="color: #B452CD">0</span>,<span style="color: #B452CD">0</span>] = np.sum(x.T<span style="color: #707a7c">@x</span>)/(n-<span style="color: #B452CD">1.0</span>)
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@@ -1014,6 +1014,7 @@ Note that the function <b>multivariate</b> returns also the covariance discussed
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<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
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<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
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<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">pandas</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">pd</span>
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<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
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<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">IPython.display</span> <span style="color: #8B008B; font-weight: bold">import</span> display
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n = <span style="color: #B452CD">10000</span>
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mean = (-<span style="color: #B452CD">1</span>, <span style="color: #B452CD">2</span>)
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@@ -1084,8 +1085,8 @@ Our own code here is not very elegant and asks for improvements.
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
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<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span><span style="color: #228B22"># extract the relevant columns from the centered design matrix of dim n x 2 x = X_centered[:,[0]]</span>
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y = X_centered[:,[<span style="color: #B452CD">1</span>]]
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<div class="highlight" style="background: #eeeedd"><pre style="line-height: 125%"><span></span><span style="color: #228B22"># extract the relevant columns from the centered design matrix of dim n x 2 x = X_centered[:,0]</span>
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y = X_centered[:,<span style="color: #B452CD">1</span>]
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Cov = np.zeros((<span style="color: #B452CD">2</span>,<span style="color: #B452CD">2</span>))
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Cov[<span style="color: #B452CD">0</span>,<span style="color: #B452CD">1</span>] = np.sum(x.T<span style="color: #707a7c">@y</span>)/(n-<span style="color: #B452CD">1.0</span>)
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Cov[<span style="color: #B452CD">0</span>,<span style="color: #B452CD">0</span>] = np.sum(x.T<span style="color: #707a7c">@x</span>)/(n-<span style="color: #B452CD">1.0</span>)
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@@ -1019,6 +1019,7 @@ Note that the function <b>multivariate</b> returns also the covariance discussed
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">IPython.display</span> <span style="color: #008000; font-weight: bold">import</span> display
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n <span style="color: #666666">=</span> <span style="color: #666666">10000</span>
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mean <span style="color: #666666">=</span> (<span style="color: #666666">-1</span>, <span style="color: #666666">2</span>)
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@@ -1089,8 +1090,8 @@ Our own code here is not very elegant and asks for improvements.
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># extract the relevant columns from the centered design matrix of dim n x 2 x = X_centered[:,[0]]</span>
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y <span style="color: #666666">=</span> X_centered[:,[<span style="color: #666666">1</span>]]
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># extract the relevant columns from the centered design matrix of dim n x 2 x = X_centered[:,0]</span>
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y <span style="color: #666666">=</span> X_centered[:,<span style="color: #666666">1</span>]
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Cov <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #666666">2</span>,<span style="color: #666666">2</span>))
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Cov[<span style="color: #666666">0</span>,<span style="color: #666666">1</span>] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(x<span style="color: #666666">.</span>T<span style="color: #AA22FF">@y</span>)<span style="color: #666666">/</span>(n<span style="color: #666666">-1.0</span>)
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Cov[<span style="color: #666666">0</span>,<span style="color: #666666">0</span>] <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sum(x<span style="color: #666666">.</span>T<span style="color: #AA22FF">@x</span>)<span style="color: #666666">/</span>(n<span style="color: #666666">-1.0</span>)
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@@ -1107,6 +1107,7 @@
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"source": [
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"import numpy as np\n",
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"import pandas as pd\n",
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"import matplotlib as plt\n",
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"from IPython.display import display\n",
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"n = 10000\n",
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"mean = (-1, 2)\n",
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@@ -1239,8 +1240,8 @@
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},
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"outputs": [],
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"source": [
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"# extract the relevant columns from the centered design matrix of dim n x 2 x = X_centered[:,[0]]\n",
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"y = X_centered[:,[1]]\n",
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"# extract the relevant columns from the centered design matrix of dim n x 2 x = X_centered[:,0]\n",
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"y = X_centered[:,1]\n",
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"Cov = np.zeros((2,2))\n",
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"Cov[0,1] = np.sum(x.T@y)/(n-1.0)\n",
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"Cov[0,0] = np.sum(x.T@x)/(n-1.0)\n",
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@@ -781,6 +781,7 @@ Note that the function _multivariate_ returns also the covariance discussed abov
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!bc pycod
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import numpy as np
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import pandas as pd
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import matplotlib as plt
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from IPython.display import display
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n = 10000
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mean = (-1, 2)
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@@ -843,8 +844,8 @@ print(np.cov(X_centered.T))
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Note that the way we define the covariance matrix here has a factor $n-1$ instead of $n$.
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Our own code here is not very elegant and asks for improvements.
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!bc pycod
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# extract the relevant columns from the centered design matrix of dim n x 2 x = X_centered[:,[0]]
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y = X_centered[:,[1]]
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# extract the relevant columns from the centered design matrix of dim n x 2 x = X_centered[:,0]
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y = X_centered[:,1]
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Cov = np.zeros((2,2))
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Cov[0,1] = np.sum(x.T@y)/(n-1.0)
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Cov[0,0] = np.sum(x.T@x)/(n-1.0)
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