Of Coward Flaws & Power Laws

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import pandas as pd
import numpy as np
from tqdm import tqdm
import os
import sys
sys.path.append("../python")
import general
import visualizations
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from IPython.display import HTML

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<form action="javascript:code_toggle()"><input type="submit" value="Click here to toggle on/off the raw code."></form>''')
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params = {'OUTPUT' : {'path' : os.path.join('output_html', 'power_law'),
                      'name' : 'power_law_20200917'},
          
          # Simulating heights of American Women
          'GAUSSIAN': {'mean' : 65, 
                       'stdev' : 3.5,
                       'n' : 10000},
          
          # Simulating batting averages (baseball)
          'BINOMIAL' : {'at_bats_per_game' : 4,
                        'batting_average' : 0.3,
                        'number_of_games' : 100000},
          
          # Value of founded company
          'POWERLAW' : {'shape' : 0.8,
                        'n' : 50000000,
                        'iteration_size' : 100000} # Recalculate mean every, say 100,000th sample
         }

Statistics, in a nutshell, is a tool for comprehending
The known and the uncertain, via samples never-ending
Ideally, with sufficient size, we soon achieve convergence,
From which, debates are settled via wisdom’s swift emergence

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random_sample = np.random.normal(params['GAUSSIAN']['mean'], params['GAUSSIAN']['stdev'], params['GAUSSIAN']['n'])
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visualizations.gaussianHistogram(random_sample)
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gaussian_means = []
for i in tqdm(range(1,params['GAUSSIAN']['n']+1)):
    gaussian_means.append(np.mean(random_sample[:i]))
100%|██████████| 10000/10000 [00:00<00:00, 35367.48it/s]
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visualizations.gaussianConvergenceLine(gaussian_means, params['GAUSSIAN']['mean'])

Canonical examples, which a lecturer deploys
Discuss the dull dimensions of two samples - girls and boys
And thus cliches perpetuate and students doze from boredom
“Deliver better content!” thus the teacher’s class implored him…

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random_sample = np.random.binomial(params['BINOMIAL']['at_bats_per_game'], params['BINOMIAL']['batting_average'], 
                                   params['BINOMIAL']['number_of_games'])
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visualizations.binomialHistogram(random_sample)
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binomial_means = []
for i in tqdm(range(1,params['BINOMIAL']['number_of_games']+1)):
    binomial_means.append(np.mean(random_sample[:i])/(params['BINOMIAL']['at_bats_per_game']))
100%|██████████| 100000/100000 [00:11<00:00, 9048.29it/s]

And though, for some, athletic stats are somewhat more compelling,
The fallacies of gamesmen oversimplify foretelling.
Because, alas, such cases of predictable behavior,
Mislead the intuition that large samples serve as savior.

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visualizations.binomialConvergenceLine(binomial_means, params['BINOMIAL']['batting_average'])