In [45]:
from collections import Counter
import math, random
In [46]:
def random_kid():
return random.choice(["boy", "girl"])
In [47]:
kid_test_list = [random_kid() for i in range(10)]
kid_test_list #random_kid 함수는 boy와 girl 두개의 값중에 하는 랜덤하게 추출함
Out[47]:
In [48]:
both_girls = 0
older_girl = 0
either_girl = 0
random.seed(0)
for _ in range(10000):
younger = random_kid()
older = random_kid()
if older == "girl": # 큰 아이가 여자일 경우 +1
older_girl += 1
if older == "girl" and younger == "girl": #둘다 여자일 경우 +1
both_girls += 1
if older == "girl" or younger == "girl": #둘중에 하나라도 여자일경우 +1
either_girl += 1
print ("P(both | older):", both_girls / older_girl) # 0.514 ~ 1/2 #큰 아이가 딸이고 둘다 딸일 확률 ()
print ("P(both | either): ", both_girls / either_girl) # 0.342 ~ 1/3 # 둘중에 한명이 딸이면서 둘 따 딸일 확률
In [49]:
#균등분포에 대한 확률밀도함수
def uniform_pdf(x):
return 1 if x >= 0 and x < 1 else 0
In [50]:
#누적분포함수
def uniform_cdf(x):
#균등분포를 따르는 확률변수의 값이 x보다 작거나 같을 확률을 반환
"returns the probability that a uniform random variable is less than x"
if x < 0:
return 0 # uniform random is never less than 0
elif x < 1:
return x # e.g. P(X < 0.4) = 0.4
else:
return 1 # uniform random is always less than 1
In [51]:
import numpy as np
x = np.arange(-1.0, 2.0, 0.1)
result_array = np.vectorize(uniform_cdf, otypes=[np.float])(x)
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x
Out[52]:
In [53]:
result_array
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In [54]:
import matplotlib.pyplot as plt
%pylab inline
plt.plot(x, result_array)
plt.axis([-1, 2, -1, 1.5])
plt.show()
In [55]:
#정규분포의 밀도함수
def normal_pdf(x, mu=0, sigma=1):
sqrt_two_pi = math.sqrt(2 * math.pi)
return (math.exp(-(x-mu) ** 2 / 2 / sigma ** 2) / (sqrt_two_pi * sigma))
#sigma는 표준편차
for sigma_value in [1,2,0.5,1]:
x = np.arange(-6.0, 6.0, 0.1)
result_array = np.vectorize(normal_pdf, otypes=[np.float])(x, sigma=sigma_value)
# plt.plot(x, result_array, "ro")
plt.plot(x, result_array)
plt.axis([-6, 6, 0, 1])
plt.show()
In [56]:
def plot_normal_pdfs(plt):
xs = [x / 10.0 for x in range(-50, 50)]
plt.plot(xs,[normal_pdf(x,sigma=1) for x in xs],'-',label='mu=0,sigma=1')
plt.plot(xs,[normal_pdf(x,sigma=0.5) for x in xs],':',label='mu=0,sigma=0.5')
plt.plot(xs,[normal_pdf(x,mu=-1) for x in xs],'-.',label='mu=-1,sigma=1')
plt.legend()
plt.show()
import matplotlib.pyplot as plt
plot_normal_pdfs(plt)
In [57]:
def normal_cdf(x, mu=0,sigma=1):
return (1 + math.erf((x - mu) / math.sqrt(2) / sigma)) / 2
In [58]:
def plot_normal_cdfs(plt):
xs = [x / 10.0 for x in range(-50, 50)]
plt.plot(xs,[normal_cdf(x,sigma=1) for x in xs],'-',label='mu=0,sigma=1')
plt.plot(xs,[normal_cdf(x,sigma=2) for x in xs],'--',label='mu=0,sigma=2')
plt.plot(xs,[normal_cdf(x,sigma=0.5) for x in xs],':',label='mu=0,sigma=0.5')
plt.plot(xs,[normal_cdf(x,mu=-1) for x in xs],'-.',label='mu=-1,sigma=1')
plt.legend(loc=4) # bottom right
plt.show()
import matplotlib.pyplot as plt
plot_normal_cdfs(plt)
In [59]:
#누적분포함수의 역함수
def inverse_normal_cdf(p, mu=0, sigma=1, tolerance=0.00001):
"""find approximate inverse using binary search"""
#이진검색을 사용해서 역함수를 근사
# if not standard, compute standard and rescale
#표준정규분포가 아니라면 표준정규분포로 변환
if mu != 0 or sigma != 1:
return mu + sigma * inverse_normal_cdf(p, tolerance=tolerance)
low_z, low_p = -10.0, 0 # normal_cdf(-10) is (very close to) 0
hi_z, hi_p = 10.0, 1 # normal_cdf(10) is (very close to) 1
while hi_z - low_z > tolerance:
mid_z = (low_z + hi_z) / 2 # consider the midpoint
mid_p = normal_cdf(mid_z) # and the cdf's value there
if mid_p < p:
# midpoint is still too low, search above it
low_z, low_p = mid_z, mid_p
elif mid_p > p:
# midpoint is still too high, search below it
hi_z, hi_p = mid_z, mid_p
else:
break
return mid_z
In [60]:
np.vectorize(inverse_normal_cdf, otypes=[np.float])([0, 0.5, 0.90, 0.95, 0.975, 1])
# 0%, 50%, 90%, 95%, 97.5%, 100%의 확률일경우 누적분포의 확률변수값
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In [61]:
# 베르누이 확률변수 (베르누이 시행 https://ko.wikipedia.org/wiki/베르누이_시행)
def bernoulli_trial(p):
return 1 if random.random() < p else 0
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# 이항 확률변수 => 단순히 n개의 독립적인 베르누이 확률변수를 더한 것
def binomial(p, n):
return sum(bernoulli_trial(p) for _ in range(n))
In [63]:
def make_hist(p, n, num_points):
data = [binomial(p, n) for _ in range(num_points)]
# use a bar chart to show the actual binomial samples
# 이항분포의 표본을 막대 그래프로 표현
histogram = Counter(data)
plt.bar([x - 0.4 for x in histogram.keys()],
[v / num_points for v in histogram.values()],
0.8,
color='0.75')
mu = p * n
sigma = math.sqrt(n * p * (1 - p))
# use a line chart to show the normal approximation
# 근사된 정규분포를 라인 차트로 표현
xs = range(min(data), max(data) + 1)
ys = [normal_cdf(i + 0.5, mu, sigma) - normal_cdf(i - 0.5, mu, sigma)
for i in xs]
plt.plot(xs,ys)
plt.show()
make_hist(0.75,100,1000)
In [64]:
make_hist(0.50,100,1000)