**The problem**: Given two random samples, s1 and s2, of size k over two disjoint populations, p1 and p2, how to combine the two k-sized random samples into one k-sized random sample over p1 ∪ p2?

**The solution**: k times, draw an element s1 ∪ s2; with probability d1 = |p1| / |p1 ∪ p2|, draw the next element from p1; with probability d2 = 1 - d1 draw the next element from p2.

(the solution was on stackoverflow)

In python:

import random import numpy # sizes e1 = 1000 e2 = 1000000 # populations p1 = xrange(e1) p2 = xrange(e1, e2) # sample size k = 500 # random samples s1 = random.sample(p1, k) s2 = random.sample(p2, k) # merge samples merge = [] for i in range(k): if s1 and s2: merge.append(s1.pop() if random.random < len(p1) / float(len(p1)+len(p2)) else s2.pop()) elif s1: merge.append(s1.pop()) else: merge.append(s2.pop()) # Validate hist = numpy.histogram(merge, bins=[0,500000,1000000]) # The two bins should be roughly equal, i.e. the error should be small. print abs(hist[0][0] - hist[0][1]) / float(k) # alternatively, use filter to count values below 500K print abs(len(filter(lambda x: x<500000, merge)) - 250) / 500.0 |