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In [1]:

    
%matplotlib inline
import numpy as np
import pandas as pd



In [2]:

    
import matplotlib
#import seaborn
#seaborn.set()



In [3]:

    
def maxn_sample(n):

    return max(np.random.random(n))



In [4]:

    
N = 10000
n = 3
df = pd.DataFrame(dict(data=[maxn_sample(n) for x in range(N)]))



In [5]:

    
df.data.plot(kind='hist')









    





<matplotlib.figure.Figure at 0xaffcf76c>






    Out[5]:





<matplotlib.axes._subplots.AxesSubplot at 0xaffcffec>



In [6]:

    
df['sample'] = np.random.random(N)
df['three'] = df.sample ** (1/n)



In [7]:

    
df.plot(kind='hist', histtype='step')









    





<matplotlib.figure.Figure at 0xaffff54c>






    Out[7]:





<matplotlib.axes._subplots.AxesSubplot at 0xaffe8c8c>

Theorem:

If $x^n$ is a uniform random variate, then distribution of x is the same as $max(a_1,...,a_n) where $a_i$ are uniform variates.



In [8]:

    
import collections



In [9]:

    
import sys
sys.path.append("/home/jng/devel")



In [10]:

    
from everest import ladybower



In [11]:

    
data = [(x+1, x) for x in range(100)]
#data = [(1 + (x//33.333), x) for x in range(100)]
res = ladybower.WeightedReservoir(data)



In [12]:

    
res.isample_without_replacement(1)









    Out[12]:





array([52])



In [13]:

    
counter = collections.Counter()

for trial in range(10000):
    sample = res.isample_without_replacement(10)

    counter.update(sample)



In [14]:

    
from matplotlib import pyplot



In [15]:

    
keys = counter.keys()
values = [counter[x] for x in keys]



In [16]:

    
sum(values), sum(range(1, 101))









    Out[16]:





(100000, 5050)



In [17]:

    
keys = [x for x in keys]



In [18]:

    
pyplot.scatter(keys, values)









    





<matplotlib.figure.Figure at 0xaff3d24c>






    Out[18]:





<matplotlib.collections.PathCollection at 0xafe5292c>



In [18]: