

In [31]:

    
%matplotlib notebook

import numpy as np
import seaborn as sns

sns.set_context("paper")

Law Of Large Numbers

?? Sample mean converges to population mean for number of samples tending to infinite
?? Average of results approximately equal to expected value (mean)
"As more observations are collected, the proportion $p_n$ of occurrences with a particular outcome converges to the probability $p$ of that outcome." OpenIntro Statistics



In [33]:

    
# Define info for die population
die = np.arange(6)+1
die_dist = np.array([1/len(values)]*len(values))



In [22]:

    
# Expected value
sum([v*p for v, p in zip(die, die_dist)])









    Out[22]:





3.5



In [11]:

    
# Simulate N die rolls
num_rolls = 10000
rolls_res = np.random.choice(die, num_rolls, p=die_dist)



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sns.plt.plot(np.arange(1, num_rolls), [rolls_res[:i].mean() for i in range(1, num_rolls)])
sns.plt.xscale('log')
sns.plt.show()

Central Limit Theorem

?? The sum of values draw independently from a distribution approximate a normal distribution the more values we extract (regardless of the underline distribution)
"if we collect a large enough sample from a population, the sample mean should be equal to, more or less, the population mean"



In [35]:

    
sns.barplot(die, die_dist)
sns.plt.show()



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np.random.choice(die, (5, 3), p=die_dist)









    Out[70]:





array([[2, 3, 1],
       [1, 1, 4],
       [6, 2, 5],
       [6, 5, 1],
       [2, 1, 3]])



In [76]:

    
num_rolls = 1000
num_dice = [1, 2, 3, 4, 5]

fig, axes = sns.plt.subplots(len(num_dice))
fig.tight_layout()
for i, num in enumerate(num_dice):
    rolls_res = np.random.choice(die, (num_rolls, num), p=die_dist).sum(axis=1)
    sns.distplot(rolls_res, ax=axes[i])
    #axes[i].set_xticklabels(axes[i].xaxis.get_majorticklabels(), rotation=30)
    #axes[i].set_xlabel(name)
sns.plt.show()

Experiment: Sum Of N Dice



In [77]:

    
# Define info for die population
die = np.arange(6)+1
die_dist = np.array([1/len(values)]*len(values))



In [83]:

    
rolls_res = np.random.choice(die, (10000, 5), p=die_dist).sum(axis=1)









    Out[83]:





array([17, 25, 17, ..., 15, 21, 10])



In [86]:

    
vals, counts = np.unique(rolls_res, return_counts=True)
num_vals = len(vals)
total_count = counts.sum()



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sns.barplot(vals, counts/total_count)
sns.plt.show()



In [92]:

    
sns.plt.plot(vals, counts/total_count)
sns.plt.show()



In [103]:

    
data = counts/total_count
p = np.arange(26) / float(26)
sns.plt.plot(vals, np.cumsum(data))
sns.plt.show()



In [ ]:

Table of Contents

Law Of Large Numbers

Central Limit Theorem

Experiment: Sum Of N Dice