random() * random() vs random()**2

Jerry Huang

May 26, 2023

Inspired by a comment on HN, the difference between random() * random() and random() ** 2.

from random import random
from collections import Counter

import matplotlib.pyplot as plt
import numpy as np

def generate_count(f, iter, precision):
  counter = Counter()
  for i in range(iter):
    counter[int(f() * precision) / precision] += 1
  return counter

random_random = generate_count(lambda: random() * random(), 1000000, 100000)

data = np.array(sorted(random_random.items()))
x, y = data.T
plt.scatter(x,y)
plt.show()

random_squared = generate_count(lambda: random() ** 2, 1000000, 100000)

data = np.array(sorted(random_squared.items()))
x, y = data.T
plt.scatter(x,y)
plt.show()

Looks like random() ** 2 clearly has much smaller values. Here's my (maybe wrong) intuition. Consider the $x\in[0,1],y\in [0,1]$ square. Think of random() * random() drawing uniformly within the 1 by 1 square, and mulitplying the $x$ and $y$ value. Then, random() ** 2 is like the diagonal line, where $x = y$. It makes sense that small values are a larger part of of the diagonal line than the square. Say $x,y \leq 0.1$ produces a value of $xy = 0$ after the precision cutoff. Then $P(xy = 0) = 0.1$ for the line, but $P(xy = 0) = 0.1^2$ for the square produces $0$.