Created
May 6, 2021 16:56
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Quantiles of the binomial and normal distributions
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#!/usr/bin/env python3 | |
# SPDX-License-Identifier: CC0-1.0 | |
from math import exp, factorial, pi, sqrt | |
def binom(n, k): | |
if n - k < 0 or n + k < 0: | |
return 0 | |
assert (n - k) % 2 == 0 and (n + k) % 2 == 0 | |
return factorial(n) / (2**n * factorial((n-k)//2) * factorial((n+k)//2)) | |
def gauss(n, k): | |
# That the normalization actually agrees with de Moivre--Laplace is | |
# easiest to verify by looking at the maximum (2 / sqrt(2*pi*n) here, | |
# 1 / sqrt(2*pi*n*p*q) there); the exponent is clear from the variance. | |
return 2 * exp(-k**2 / (2*n)) / sqrt(2*pi*n) | |
# <https://en.wikipedia.org/wiki/68-95-99.7_rule> | |
intervals = [(0.5, 0.382_924_922_548_026), | |
(1.0, 0.682_689_492_137_086), | |
(1.5, 0.866_385_597_462_284), | |
(2.0, 0.954_499_736_103_642), | |
(2.5, 0.987_580_669_348_448), | |
(3.0, 0.997_300_203_936_740), | |
(3.5, 0.999_534_741_841_929), | |
(4.0, 0.999_936_657_516_334), | |
(4.5, 0.999_993_204_653_751), | |
(5.0, 0.999_999_426_696_856)] | |
n = 100 | |
ks = list(range(-n, n+1, 2)) | |
sigma = sqrt(n) | |
print("SCORE \tEXACT \tNORMAL \tINTEGRAL ") | |
for score, integral in intervals: | |
exact = sum(binom(n, k) for k in ks if abs(k) <= score * sigma) | |
normal = sum(gauss(n, k) for k in ks if abs(k) <= score * sigma) | |
print("{:.1f} sigma\t{:.10f}\t{:.10f}\t{:.10f}\t" | |
.format(score, exact, normal, integral)) |
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