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@@ -7,8 +7,8 @@
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<title>1. Elements of Probability Theory and Statistical Data Analysis &#8212; Applied Data Analysis and Machine Learning</title>
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About the course
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@@ -116,7 +122,7 @@ const thebe_selector_output = ".output, .cell_output"
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Review of Statistics with Resampling Techniques and Linear Algebra
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@@ -133,7 +139,7 @@ const thebe_selector_output = ".output, .cell_output"
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From Regression to Support Vector Machines
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@@ -170,7 +176,7 @@ const thebe_selector_output = ".output, .cell_output"
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Decision Trees, Ensemble Methods and Boosting
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Dimensionality Reduction
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@@ -204,7 +210,7 @@ const thebe_selector_output = ".output, .cell_output"
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Deep Learning Methods
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@@ -281,7 +287,7 @@ const thebe_selector_output = ".output, .cell_output"
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@@ -299,7 +305,7 @@ const thebe_selector_output = ".output, .cell_output"
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@@ -361,7 +367,72 @@ const thebe_selector_output = ".output, .cell_output"
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<h1>Elements of Probability Theory and Statistical Data Analysis</h1>
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<h2> Contents </h2>
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1.1. Domains and probabilities
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<a class="reference internal nav-link" href="#stochastic-variables-and-the-main-concepts-the-discrete-case">
1.1.1. Stochastic variables and the main concepts, the discrete case
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<a class="reference internal nav-link" href="#properties-of-pdfs">
1.1.2. Properties of PDFs
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<a class="reference internal nav-link" href="#expectation-values">
1.1.3. Expectation values
</a>
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<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#probability-distribution-functions">
1.1.4. Probability Distribution Functions
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<a class="reference internal nav-link" href="#meet-the-covariance">
1.1.5. Meet the covariance!
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<a class="reference internal nav-link" href="#numerical-experiments-and-the-covariance-central-limit-theorem">
1.1.6. Numerical experiments and the covariance, central limit theorem
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<a class="reference internal nav-link" href="#random-numbers">
1.1.7. Random Numbers
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1.1.8. Autocorrelation function
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@@ -845,37 +916,27 @@ uncorrelated.</p>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>2.3120271598582915
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3.79145214e-02 1.01869785e-01]
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3.39794864e+00 4.78988962e+00 6.50680321e+00 2.32750531e+00
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[1.48316523e+01 7.68176047e-02 4.90999452e+00 1.34040204e+01
9.27621662e+00 1.30761405e+01 1.77632220e+01 6.35396404e+00
6.61576236e+00 1.77754132e+01]
[1.02642055e+01 5.31614188e-02 3.39794864e+00 9.27621662e+00
6.41958102e+00 9.04930820e+00 1.22929905e+01 4.39724390e+00
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[1.44688507e+01 7.49385454e-02 4.78988962e+00 1.30761405e+01
9.04930820e+00 1.27562809e+01 1.73287103e+01 6.19853775e+00
6.45393214e+00 1.73406033e+01]
[1.96551427e+01 1.01799917e-01 6.50680321e+00 1.77632220e+01
1.22929905e+01 1.73287103e+01 2.35401056e+01 8.42037468e+00
8.76731400e+00 2.35562617e+01]
[7.03071040e+00 3.64141716e-02 2.32750531e+00 6.35396404e+00
4.39724390e+00 6.19853775e+00 8.42037468e+00 3.01199624e+00
3.13609760e+00 8.42615374e+00]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>1.1297300314822336
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14.31835835 18.32068012 16.42732954 5.1023858 ]
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9.48971452 12.14231548 10.88746795 3.38168549]
[ 7.56183518 12.6052136 12.72671218 7.18998208 8.43482628 6.14298603
6.91124889 8.84310736 7.92921648 2.46284227]
[ 8.50754416 14.18166474 14.31835835 8.08918584 9.48971452 6.91124889
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[ 3.03168642 5.05367466 5.1023858 2.8826033 3.38168549 2.46284227
2.77085375 3.54537329 3.17897671 0.98740124]]
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@@ -1143,15 +1204,15 @@ more practically oriented methods like the blocking technique.</p>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.11674401539815678
4.220506557673408
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[ 2.82964842 9.09440475 11.11598862]]
[21.63207808 0.07291729 2.49614357]
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>-0.09083636328656121
3.6943316601792833
-0.35793003441520066
0.8866190885623907 9.823585220707946 13.828190347382744
2.7923086060375724 2.5472246386316972 7.87667593906086
[[ 0.88661909 2.79230861 2.54722464]
[ 2.79230861 9.82358522 7.87667594]
[ 2.54722464 7.87667594 13.82819035]]
[20.65578316 0.07532297 3.80728853]
</pre></div>
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@@ -1481,7 +1542,7 @@ assumption for approximating <span class="math notranslate nohighlight">\(\sigma
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>-0.01212754811385597 1.0209518407597062
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.008818897251043893 0.9940672992288855
</pre></div>
</div>
<img alt="_images/statistics_188_1.png" src="_images/statistics_188_1.png" />
@@ -1515,54 +1576,42 @@ For the remaining values we notice that there are still non-zero values for the
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