연속 확률 분포:
1) 확률 밀도 함수(Probability Density Function: PDF)
특정 확률 변수 구간을 적분한 값으로 확률을 계산할 수 있는 함수
P(a <= x < b) = Integral from a to b [pdf(x)dx]
2) 누적 분포 함수(Cumulative Distribution Function: CDF)
확률 변수의 값이 특정 값보다 작거나 같을 확률을 나타내는 함수
cdf(x) = P(x <= b)
P(a <= x < b) = P(x < b) - P(x <= a)
import matplotlib.pyplot as plt
def uniform_pdf(x):
""" Uniform Distribution(균등 분포) 확률 밀도 함수 """
return 1 if 0 <= x < 1 else 0
def uniform_cdf(x):
""" 균등 분포 누적 분포 함수 """
if x < 0:
return 0
elif 0 <= x < 1:
return x
else:
return 1
.
.
if __name__ == '__main__':
# -2 <= x <= 2 구간을 0.1씩 나눔
xs = [x / 10 for x in range(-20, 21)]
print(xs)
[-2.0, -1.9, -1.8, -1.7, -1.6, -1.5, -1.4, -1.3, -1.2, -1.1, -1.0, -0.9, -0.8, -0.7, -0.6, -0.5, -0.4, -0.3, -0.2, -0.1, 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0]
ys = [uniform_pdf(x) for x in xs]
print(ys)
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
plt.plot(xs, ys)
plt.title('Uniform distribution PDF')
plt.show()
ys2 = [uniform_cdf(x) for x in xs]
plt.plot(xs, ys2)
plt.title('Uniform distribution CDF')
plt.show()
def normal_pdf(x, mu=0.0, sigma=1.0):
"""평균이 mu이고, 표준편차가 sigma인 정규 분포(Normal Distribution)
확률 밀도 함수
"""
return math.exp(-(x - mu) ** 2 / (2 * sigma ** 2)) / (SQRT_TWO_PI * sigma)
위의 리턴값은 다음의 공식을 코딩한 것이다.
# mu = 0.0, sigma = 2.0
ys2 = [normal_pdf(x, sigma = 2.0) for x in xs]
ys3 = [normal_pdf(x, sigma=0.5) for x in xs]
plt.plot(xs, ys1, '-', label = 'mu=0, sigma=1')
plt.plot(xs, ys2, '--', label = 'mu=0, sigma=2')
plt.plot(xs, ys3, ':', label='mu=0, sigma=0.5')
plt.legend()
plt.title('Normal Distribution PDF')
plt.show()
# mu = -1.0, sigma = 1.0
ys4 = [normal_pdf(x, mu=-1.0) for x in xs]
plt.plot(xs, ys1, '-', label='mu=0, sigma=1')
plt.plot(xs, ys4, '-.', label='mu=-1, sigma=1.0')
plt.legend()
plt.title('Normal Distribution PDF')
plt.show()
def normal_cdf(x, mu=0.0, sigma=1.0):
"""평균이 mu이고, 표준편차가 sigma인
정규 분포(Normal Distribution)의 누적 분포 함수(Cumulative Distribution Function)
math.erf() 함수(error function)를 이용해서 구현"""
return (1 + math.erf((x - mu) / (math.sqrt(2) * sigma))) / 2
# mu=0, sigma=1인 CDF
ys1 = [normal_cdf(x) for x in xs]
plt.plot(xs, ys1, '-', label='mu=0, sigma=1')
plt.legend()
plt.title('Normal Distribution CDF')
plt.show()
# mu=0, sigma=1인 CDF
ys1 = [normal_cdf(x) for x in xs]
# mu=0, sigma=2인 CDF
ys2 = [normal_cdf(x, sigma=2.0) for x in xs]
# mu=0, sigma=0.5인 CDF
ys3 = [normal_cdf(x, sigma=0.5) for x in xs]
plt.plot(xs, ys1, '-', label='mu=0, sigma=1')
plt.plot(xs, ys2, '--', label='mu=0, sigma=2')
plt.plot(xs, ys3, ':', label='mu=0, sigma=0.5')
plt.legend()
plt.title('Normal Distribution CDF')
plt.show()
# mu=0, sigma=1인 CDF
ys1 = [normal_cdf(x) for x in xs]
# mu=-1, sigma=1인 CDF
ys4 = [normal_cdf(x, mu=-1) for x in xs]
plt.plot(xs, ys1, '-', label='mu=0, sigma=1')
plt.plot(xs, ys4, '-.', label='mu=-1, sigma=1')
plt.legend()
plt.title('Normal Distribution CDF')
plt.show()
def inverse_normal_cdf(p, mu=0.0, sigma=1.0, tolerance=0.00001):
"""누적 확률 p를 알고 있을 때 정규 분포 확률 변수 x = ?"""
# 표준 정규 분포가 아니라면 표준 정규 분포로 변환
if mu != 0.0 or sigma != 1.0:
return mu + sigma * inverse_normal_cdf(p, tolerance=tolerance)
low_z = -10.0 # 하한(lower bound)
high_z = 10.0 # 상한(upper bound)
while high_z - low_z > tolerance:
mid_z = (low_z + high_z) / 2.0 # 중간 값
mid_p = normal_cdf(mid_z) # 중간 값에서의 누적 확률
if mid_p < p:
low_z = mid_z
else:
high_z = mid_z
return mid_z
# 누적 확률이 0.9, 0.99, 0.999인 확률 변수 x를 찾음.
# 표준 정규 분포표와 비교
x1 = inverse_normal_cdf(0.9)
print('x1 =', x1)
x2 = inverse_normal_cdf(0.99)
print('x2 =', x2)
x3 = inverse_normal_cdf(0.999)
print('x3 =', x3)
x1 = 1.2815570831298828
x2 = 2.326345443725586
x3 = 3.090238571166992
