확률 및 분포

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하룡 석
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1 확률및분포 박창이 서울시립대학교통계학과 박창이 ( 서울시립대학교통계학과 ) 확률및분포 1 / 15

2 학습내용 조건부확률막대그래프히스토그램선그래프산점도참고 박창이 ( 서울시립대학교통계학과 ) 확률및분포 2 / 15

3 조건부확률 I 첫째가딸일때두아이모두딸일확률 (1/2) 과둘중의하나가딸일때둘다딸일확률 (1/3) 에대한모의실험 >>> from collections import Counter >>> import math, random >>> from matplotlib import pyplot as plt >>> def random_kid(): return random.choice(["boy", "girl"]) >>> 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": older_girl += 1 박창이 ( 서울시립대학교통계학과 ) 확률및분포 3 / 15

4 조건부확률 II if older == "girl" and younger == "girl": both_girls += 1 if older == "girl" or younger == "girl": either_girl += 1 >>> print("p(both older):", both_girls / older_girl) # ~ 1/2 P(both older): >>> print("p(both either): ", both_girls / either_girl) # ~ 1/3 P(both either): 박창이 ( 서울시립대학교통계학과 ) 확률및분포 4 / 15

5 확률분포 I 균일분포 >>> def uniform_pdf(x): return 1 if x >= 0 and x < 1 else 0 >>> def uniform_cdf(x): "returns the probability that a uniform random variable is less tha 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 박창이 ( 서울시립대학교통계학과 ) 확률및분포 5 / 15

6 확률분포 II 정규분포확률밀도함수 >>> 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)) >>> 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=2) for x in xs], --, label= mu=0,sigma=2 ) >>> 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(loc=4) # bottom right >>> plt.show() 박창이 ( 서울시립대학교통계학과 ) 확률및분포 6 / 15

7 확률분포 III 박창이 ( 서울시립대학교통계학과 ) 확률및분포 7 / 15

8 확률분포 IV 분포함수 >>> def normal_cdf(x, mu=0,sigma=1): return (1 + math.erf((x - mu) / math.sqrt(2) / sigma)) / 2 >>> 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() 박창이 ( 서울시립대학교통계학과 ) 확률및분포 8 / 15

9 확률분포 V 박창이 ( 서울시립대학교통계학과 ) 확률및분포 9 / 15

10 확률분포 VI 분위수함수 >>> def inverse_normal_cdf(p, mu=0, sigma=1, tolerance= ): """find approximate inverse using binary search""" # if not standard, compute standard and rescale 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) ~ 0 hi_z, hi_p = 10.0, 1 # normal_cdf(10) ~ 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 박창이 ( 서울시립대학교통계학과 ) 확률및분포 10 / 15

11 확률분포 VII 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 박창이 ( 서울시립대학교통계학과 ) 확률및분포 11 / 15

12 중심극한정리 I >>> def bernoulli_trial(p): return 1 if random.random() < p else 0 >>> def binomial(p, n): return sum(bernoulli_trial(p) for _ in range(n)) >>> 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 histogram = Counter(data) plt.bar([x for x in histogram.keys()], [v / num_points for v in histogram.values()], 0.8, color= 0.75 ) 박창이 ( 서울시립대학교통계학과 ) 확률및분포 12 / 15

13 중심극한정리 II mu = p * n sigma = math.sqrt(n * p * (1 - p)) # use a line chart to show the normal approximation 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, 10000) 박창이 ( 서울시립대학교통계학과 ) 확률및분포 13 / 15

14 중심극한정리 III 박창이 ( 서울시립대학교통계학과 ) 확률및분포 14 / 15

15 참고 scipy.stats: 여러가지확률분포의확률밀도함수및누적분포함수구현됨 박창이 ( 서울시립대학교통계학과 ) 확률및분포 15 / 15

데이터 시각화

데이터 시각화 데이터시각화 박창이 서울시립대학교통계학과 박창이 ( 서울시립대학교통계학과 ) 데이터시각화 1 / 22 학습내용 matplotlib 막대그래프히스토그램선그래프산점도참고 박창이 ( 서울시립대학교통계학과 ) 데이터시각화 2 / 22 matplotlib I 간단한막대그래프, 선그래프, 산점도등을그릴때유용 http://matplotlib.org 에서설치방법참고윈도우의경우명령프롬프트를관리자권한으로실행한후아래의코드실행