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    <p>Weights define a probability distribution function (pdf). Random numbers from any such pdf can be generated by <a href="http://en.wikipedia.org/wiki/Pseudorandom_number_generator#Non-uniform_generators" rel="nofollow noreferrer">applying its associated inverse cumulative distribution function</a> to uniform random numbers between 0 and 1.</p> <p>See also this <a href="https://stackoverflow.com/questions/977354/generating-non-uniform-random-numbers/977508#977508">SO explanation</a>, or, as explained by <a href="http://en.wikipedia.org/wiki/Cumulative%5Fdistribution%5Ffunction#Inverse" rel="nofollow noreferrer">Wikipedia</a>:</p> <blockquote> <p>If Y has a U[0,1] distribution then F⁻¹(Y) is distributed as F. This is used in random number generation using the inverse transform sampling-method.</p> </blockquote> <pre><code>import random import bisect import collections def cdf(weights): total = sum(weights) result = [] cumsum = 0 for w in weights: cumsum += w result.append(cumsum / total) return result def choice(population, weights): assert len(population) == len(weights) cdf_vals = cdf(weights) x = random.random() idx = bisect.bisect(cdf_vals, x) return population[idx] weights=[0.3, 0.4, 0.3] population = 'ABC' counts = collections.defaultdict(int) for i in range(10000): counts[choice(population, weights)] += 1 print(counts) # % test.py # defaultdict(&lt;type 'int'&gt;, {'A': 3066, 'C': 2964, 'B': 3970}) </code></pre> <p>The <code>choice</code> function above uses <code>bisect.bisect</code>, so selection of a weighted random variable is done in <code>O(log n)</code> where <code>n</code> is the length of <code>weights</code>. </p> <hr> <p>Note that as of version 1.7.0, NumPy has a Cythonized <a href="http://docs.scipy.org/doc/numpy-dev/reference/generated/numpy.random.choice.html" rel="nofollow noreferrer">np.random.choice function</a>. For example, this generates 1000 samples from the population <code>[0,1,2,3]</code> with weights <code>[0.1, 0.2, 0.3, 0.4]</code>:</p> <pre><code>import numpy as np np.random.choice(4, 1000, p=[0.1, 0.2, 0.3, 0.4]) </code></pre> <p><code>np.random.choice</code> also has a <code>replace</code> parameter for sampling with or without replacement.</p> <hr> <p>A theoretically better algorithm is the <a href="http://en.wikipedia.org/wiki/Alias_method" rel="nofollow noreferrer">Alias Method</a>. It builds a table which requires <code>O(n)</code> time, but after that, samples can be drawn in <code>O(1)</code> time. So, if you need to draw many samples, in theory the Alias Method may be faster. There is a Python implementation of the Walker Alias Method <a href="http://code.activestate.com/recipes/576564-walkers-alias-method-for-random-objects-with-diffe/" rel="nofollow noreferrer">here</a>, and a <a href="https://gist.github.com/ntamas/1109133" rel="nofollow noreferrer">numpy version here</a>.</p>
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