def binomialvariate(self, n=1, p=0.5):
"""Binomial random variable.
Gives the number of successes for *n* independent trials
with the probability of success in each trial being *p*:
sum(random() < p for i in range(n))
Returns an integer in the range: 0 <= X <= n
"""
# Error check inputs and handle edge cases
if n < 0:
raise ValueError("n must be non-negative")
if p <= 0.0 or p >= 1.0:
if p == 0.0:
return 0
if p == 1.0:
return n
raise ValueError("p must be in the range 0.0 <= p <= 1.0")
random = self.random
# Fast path for a common case
if n == 1:
return _index(random() < p)
# Exploit symmetry to establish: p <= 0.5
if p > 0.5:
return n - self.binomialvariate(n, 1.0 - p)
if n * p < 10.0:
# BG: Geometric method by Devroye with running time of O(np).
# https://dl.acm.org/doi/pdf/10.1145/42372.42381
x = y = 0
c = _log(1.0 - p)
if not c:
return x
while True:
y += _floor(_log(random()) / c) + 1
if y > n:
return x
x += 1
# BTRS: Transformed rejection with squeeze method by Wolfgang Hörmann
# https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.47.8407&rep=rep1&type=pdf
assert n * p >= 10.0 and p <= 0.5
setup_complete = False
spq = _sqrt(n * p *
(1.0 - p)) # Standard deviation of the distribution
b = 1.15 + 2.53 * spq
a = -0.0873 + 0.0248 * b + 0.01 * p
c = n * p + 0.5
vr = 0.92 - 4.2 / b
while True:
u = random()
v = random()
u -= 0.5
us = 0.5 - _fabs(u)
k = _floor((2.0 * a / us + b) * u + c)
if k < 0 or k > n:
continue
# The early-out "squeeze" test substantially reduces
# the number of acceptance condition evaluations.
if us >= 0.07 and v <= vr:
return k
# Acceptance-rejection test.
# Note, the original paper errorneously omits the call to log(v)
# when comparing to the log of the rescaled binomial distribution.
if not setup_complete:
alpha = (2.83 + 5.1 / b) * spq
lpq = _log(p / (1.0 - p))
m = _floor((n + 1) * p) # Mode of the distribution
h = _lgamma(m + 1) + _lgamma(n - m + 1)
setup_complete = True # Only needs to be done once
v *= alpha / (a / (us * us) + b)
if _log(v) <= h - _lgamma(k + 1) - _lgamma(n - k +
1) + (k - m) * lpq:
return k
def lgamma(x):
return _lgamma(x)
def _igamc(a, x):
"""Complemented incomplete Gamma integral.
SYNOPSIS:
double a, x, y, igamc();
y = igamc( a, x );
DESCRIPTION:
The function is defined by::
igamc(a,x) = 1 - igam(a,x)
inf.
-
1 | | -t a-1
= ----- | e t dt.
- | |
| (a) -
x
In this implementation both arguments must be positive.
The integral is evaluated by either a power series or
continued fraction expansion, depending on the relative
values of a and x.
"""
# Compute x**a * exp(-x) / Gamma(a)
# TODO: Return this to math.lgamma once drop Python 2.6
ax = math.exp(a * math.log(x) - x - _lgamma(a))
# Continued fraction
y = 1.0 - a
z = x + y + 1.0
c = 0.0
pkm2 = 1.0
qkm2 = x
pkm1 = x + 1.0
qkm1 = z * x
ans = pkm1 / qkm1
while True:
c += 1.0
y += 1.0
z += 2.0
yc = y * c
pk = pkm1 * z - pkm2 * yc
qk = qkm1 * z - qkm2 * yc
if qk != 0:
r = pk / qk
t = abs((ans - r) / r)
ans = r
else:
t = 1.0
pkm2 = pkm1
pkm1 = pk
qkm2 = qkm1
qkm1 = qk
if abs(pk) > BIG:
pkm2 *= BIGINV
pkm1 *= BIGINV
qkm2 *= BIGINV
qkm1 *= BIGINV
if t <= MACHEP:
return ans * ax
def lgamma(x):
return _lgamma(x)
def _igamc(a, x):
"""Complemented incomplete Gamma integral.
SYNOPSIS:
double a, x, y, igamc();
y = igamc( a, x );
DESCRIPTION:
The function is defined by::
igamc(a,x) = 1 - igam(a,x)
inf.
-
1 | | -t a-1
= ----- | e t dt.
- | |
| (a) -
x
In this implementation both arguments must be positive.
The integral is evaluated by either a power series or
continued fraction expansion, depending on the relative
values of a and x.
"""
# Compute x**a * exp(-x) / Gamma(a)
# TODO: Return this to math.lgamma once drop Python 2.6
ax = math.exp(a * math.log(x) - x - _lgamma(a))
# Continued fraction
y = 1.0 - a
z = x + y + 1.0
c = 0.0
pkm2 = 1.0
qkm2 = x
pkm1 = x + 1.0
qkm1 = z * x
ans = pkm1 / qkm1
while True:
c += 1.0
y += 1.0
z += 2.0
yc = y * c
pk = pkm1 * z - pkm2 * yc
qk = qkm1 * z - qkm2 * yc
if qk != 0:
r = pk / qk
t = abs((ans - r) / r)
ans = r
else:
t = 1.0
pkm2 = pkm1
pkm1 = pk
qkm2 = qkm1
qkm1 = qk
if abs(pk) > BIG:
pkm2 *= BIGINV
pkm1 *= BIGINV
qkm2 *= BIGINV
qkm1 *= BIGINV
if t <= MACHEP:
return ans * ax