Dice Experiments on Varience and Expectation

docs/_data/sidebars/home_sidebar.yml

@@ -21,6 +21,9 @@ entries:
  - output: web,pdf
  title: Design Document
  url: /design_document
+ - output: web,pdf
+ title: Dice Experiments
+ url: /dice_experiments
  - output: web,pdf
  title: Function Approximation
  url: /function_approximation

docs/dice_experiments.html

@@ -0,0 +1,249 @@
+---
+
+title: Dice Experiments
+
+keywords: fastai
+sidebar: home_sidebar
+
+summary: "Probability on Dice"
+description: "Probability on Dice"
+---
+<!--
+
+#################################################
+### THIS FILE WAS AUTOGENERATED! DO NOT EDIT! ###
+#################################################
+# file to edit: notebooks/dice_experiments.ipynb
+# command to build the docs after a change: nbdev_build_docs
+
+-->
+
+<div class="container" id="notebook-container">
+
+ {% raw %}
+
+<div class="cell border-box-sizing code_cell rendered">
+
+</div>
+ {% endraw %}
+
+ {% raw %}
+
+<div class="cell border-box-sizing code_cell rendered">
+
+</div>
+ {% endraw %}
+
+<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
+<div class="text_cell_render border-box-sizing rendered_html">
+<h3 id="Standard-Deviation-and-Expectation">Standard Deviation and Expectation<a class="anchor-link" href="#Standard-Deviation-and-Expectation"> </a></h3>
+</div>
+</div>
+</div>
+<div class="cell border-box-sizing text_cell rendered"><div class="inner_cell">
+<div class="text_cell_render border-box-sizing rendered_html">
+<p>Q: If you had a 2 die (one regular and one loaded (as described by the table below)
+What is the expected value and the standard deviation of each number rolled?</p>
+<p>For the biased die:</p>
+<ul>
+<li>P(1) = 1/12 </li>
+<li>P(2) = 1/12 </li>
+<li>P(3) = 1/3 </li>
+<li>P(4) = 1/3 </li>
+<li>P(5) = 1/12 </li>
+<li>P(6) = 1/12 </li>
+</ul>
+
+</div>
+</div>
+</div>
+ {% raw %}
+
+<div class="cell border-box-sizing code_cell rendered">
+<div class="input">
+
+<div class="inner_cell">
+ <div class="input_area">
+<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">get_loaded_die_prob</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
+ <span class="k">assert</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="nb">int</span><span class="p">)</span>
+ <span class="n">probs</span> <span class="o">=</span> <span class="p">{</span>
+ <span class="mi">1</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="mi">12</span><span class="p">,</span>
+ <span class="mi">2</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="mi">12</span><span class="p">,</span>
+ <span class="mi">3</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="mi">3</span><span class="p">,</span>
+ <span class="mi">4</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="mi">3</span><span class="p">,</span>
+ <span class="mi">5</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="mi">12</span><span class="p">,</span>
+ <span class="mi">6</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="mi">12</span><span class="p">,</span>
+ <span class="p">}</span>
+ <span class="k">return</span> <span class="n">probs</span><span class="p">[</span><span class="n">x</span><span class="p">]</span>
+
+<span class="k">def</span> <span class="nf">get_fair_die_prob</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
+ <span class="k">assert</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="nb">int</span><span class="p">)</span>
+ <span class="n">probs</span> <span class="o">=</span> <span class="p">{</span>
+ <span class="mi">1</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span>
+ <span class="mi">2</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span>
+ <span class="mi">3</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span>
+ <span class="mi">4</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span>
+ <span class="mi">5</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span>
+ <span class="mi">6</span><span class="p">:</span> <span class="mi">1</span><span class="o">/</span><span class="mi">6</span><span class="p">,</span>
+ <span class="p">}</span>
+ <span class="k">return</span> <span class="n">probs</span><span class="p">[</span><span class="n">x</span><span class="p">]</span>
+
+<span class="n">fair_die_probs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([</span><span class="mi">1</span><span class="o">/</span><span class="mi">6</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">6</span><span class="p">)])</span>
+<span class="n">loaded_die_probs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">([</span><span class="mi">1</span><span class="o">/</span><span class="mi">12</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">12</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">12</span><span class="p">,</span> <span class="mi">1</span><span class="o">/</span><span class="mi">12</span><span class="p">])</span>
+<span class="n">loaded_die_probs</span><span class="p">,</span> <span class="n">fair_die_probs</span>
+</pre></div>
+
+ </div>
+</div>
+</div>
+
+<div class="output_wrapper">
+<div class="output">
+
+<div class="output_area">
+
+
+
+<div class="output_text output_subarea output_execute_result">
+<pre>(array([0.08333333, 0.08333333, 0.33333333, 0.33333333, 0.08333333,
+ 0.08333333]),
+ array([0.16666667, 0.16666667, 0.16666667, 0.16666667, 0.16666667,
+ 0.16666667]))</pre>
+</div>
+
+</div>
+
+</div>
+</div>
+
+</div>
+ {% endraw %}
+
+ {% raw %}
+
+<div class="cell border-box-sizing code_cell rendered">
+<div class="input">
+
+<div class="inner_cell">
+ <div class="input_area">
+<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">###########################</span>
+<span class="c1"># Analytical Calculation </span>
+<span class="c1">###########################</span>
+
+<span class="c1"># expected value calculations</span>
+<span class="n">fair_expected_value</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">stop</span><span class="o">=</span><span class="mi">6</span><span class="p">,</span><span class="n">num</span><span class="o">=</span><span class="mi">6</span><span class="p">),</span> <span class="n">fair_die_probs</span><span class="p">)</span>
+<span class="n">loaded_expected_value</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">stop</span><span class="o">=</span><span class="mi">6</span><span class="p">,</span><span class="n">num</span><span class="o">=</span><span class="mi">6</span><span class="p">),</span> <span class="n">loaded_die_probs</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;&quot;&quot;</span>
+<span class="s2"> expected value of fair die: </span><span class="si">{</span><span class="n">fair_expected_value</span><span class="si">}</span><span class="s2"></span>
+<span class="s2"> expected value loaded die: </span><span class="si">{</span><span class="n">loaded_expected_value</span><span class="si">}</span><span class="s2"></span>
+<span class="s2"> &quot;&quot;&quot;</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;=&quot;</span><span class="o">*</span><span class="mi">40</span><span class="p">)</span>
+<span class="c1"># varience calculations; varience = E[(x-mean)^2]</span>
+<span class="n">loaded_die_variences</span> <span class="o">=</span> <span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mi">6</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">6</span><span class="p">)</span> <span class="o">-</span> <span class="n">loaded_expected_value</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">loaded_die_probs</span>
+<span class="n">fair_die_variences</span> <span class="o">=</span> <span class="p">((</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">stop</span><span class="o">=</span><span class="mi">6</span><span class="p">,</span> <span class="n">num</span><span class="o">=</span><span class="mi">6</span><span class="p">)</span> <span class="o">-</span> <span class="n">fair_expected_value</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">fair_die_probs</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;&quot;&quot;</span>
+<span class="s2"> fair die variences: </span><span class="si">{</span><span class="n">fair_die_variences</span><span class="si">}</span><span class="s2"></span>
+<span class="s2"> loaded die variences: </span><span class="si">{</span><span class="n">loaded_die_variences</span><span class="si">}</span><span class="s2"></span>
+<span class="s2"> </span>
+<span class="s2">&quot;&quot;&quot;</span><span class="p">)</span>
+
+<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;=&quot;</span><span class="o">*</span><span class="mi">40</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;&quot;&quot; </span>
+<span class="s2">fair die:</span>
+<span class="s2"> total varience = </span><span class="si">{</span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">fair_die_variences</span><span class="p">)</span><span class="si">}</span><span class="s2"></span>
+<span class="s2"> total std dev = </span><span class="si">{</span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">fair_die_variences</span><span class="p">)</span><span class="o">**</span><span class="mf">0.5</span><span class="si">}</span><span class="s2"></span>
+
+<span class="s2">loaded die:</span>
+<span class="s2"> total varience = </span><span class="si">{</span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">loaded_die_variences</span><span class="p">)</span><span class="si">}</span><span class="s2"></span>
+<span class="s2"> total std dev = </span><span class="si">{</span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">loaded_die_variences</span><span class="p">)</span><span class="o">**</span><span class="mf">0.5</span><span class="si">}</span><span class="s2"></span>
+
+<span class="s2">&quot;&quot;&quot;</span><span class="p">)</span>
+</pre></div>
+
+ </div>
+</div>
+</div>
+
+<div class="output_wrapper">
+<div class="output">
+
+<div class="output_area">
+
+<div class="output_subarea output_stream output_stdout output_text">
+<pre>
+ expected value of fair die: 3.5
+ expected value loaded die: 3.4999999999999996
+
+========================================
+
+ fair die variences: [1.04166667 0.375 0.04166667 0.04166667 0.375 1.04166667]
+ loaded die variences: [0.52083333 0.1875 0.08333333 0.08333333 0.1875 0.52083333]
+
+
+========================================
+
+fair die:
+ total varience = 2.9166666666666665
+ total std dev = 1.707825127659933
+
+loaded die:
+ total varience = 1.5833333333333333
+ total std dev = 1.2583057392117916
+
+
+</pre>
+</div>
+</div>
+
+</div>
+</div>
+
+</div>
+ {% endraw %}
+
+ {% raw %}
+
+<div class="cell border-box-sizing code_cell rendered">
+<div class="input">
+
+<div class="inner_cell">
+ <div class="input_area">
+<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">###########################</span>
+<span class="c1"># Emperical Calculation </span>
+<span class="c1">###########################</span>
+<span class="n">NUM_TRIALS</span> <span class="o">=</span> <span class="mi">1_000_000</span>
+<span class="n">fair_emp_varience</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">stop</span><span class="o">=</span><span class="mi">6</span><span class="p">,</span><span class="n">num</span><span class="o">=</span><span class="mi">6</span><span class="p">),</span> <span class="n">p</span><span class="o">=</span><span class="n">fair_die_probs</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">NUM_TRIALS</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span>
+<span class="n">loaded_emp_varience</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">stop</span><span class="o">=</span><span class="mi">6</span><span class="p">,</span><span class="n">num</span><span class="o">=</span><span class="mi">6</span><span class="p">),</span> <span class="n">p</span><span class="o">=</span><span class="n">loaded_die_probs</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">NUM_TRIALS</span><span class="p">))</span><span class="o">**</span><span class="mi">2</span>
+</pre></div>
+
+ </div>
+</div>
+</div>
+
+</div>
+ {% endraw %}
+
+ {% raw %}
+
+<div class="cell border-box-sizing code_cell rendered">
+<div class="input">
+
+<div class="inner_cell">
+ <div class="input_area">
+<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># checksum</span>
+<span class="n">epsilon</span> <span class="o">=</span> <span class="mf">1e-2</span>
+<span class="k">assert</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">fair_die_variences</span><span class="p">)</span><span class="o">-</span><span class="n">fair_emp_varience</span> <span class="o">&lt;</span> <span class="n">epsilon</span>
+<span class="k">assert</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">loaded_die_variences</span><span class="p">)</span><span class="o">-</span><span class="n">loaded_emp_varience</span> <span class="o">&lt;</span> <span class="n">epsilon</span>
+</pre></div>
+
+ </div>
+</div>
+</div>
+
+</div>
+ {% endraw %}
+
+</div>
+
+

docs/sidebar.json

@@ -5,6 +5,7 @@
  "Custom Tensorflow Training": "/custom_tf_training",
  "Data Generator": "/data_generator",
  "Design Document": "/design_document",
+ "Dice Experiments": "/dice_experiments",
  "Function Approximation": "/function_approximation",
  "Kelly Criterion": "/kelly_criterion",
  "Transformers": "/transformers"

jovsatools/_nbdev.py

@@ -17,6 +17,7 @@
 modules = ["custom_pytorch_training.py",
  "custom_tf_training.py",
  "data_generator.py",
+ "dice_experiments.py",
  "function_approximation.py"]
 
 doc_url = "https://jovsa.github.io/jovsatools/"

jovsatools/dice_experiments.py

@@ -0,0 +1,9 @@
+# AUTOGENERATED! DO NOT EDIT! File to edit: notebooks/dice_experiments.ipynb (unless otherwise specified).
+
+__all__ = []
+
+# Cell
+from nbdev.showdoc import *
+import jovsatools
+import fastcore
+import numpy as np