
import enum, random

class Kid(enum.Enum):
    BOY = 0
    GIRL = 1

def random_kid() -> Kid:
    return random.choice([Kid.BOY, Kid.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 == Kid.GIRL:
        older_girl += 1
    if older == Kid.GIRL and younger == Kid.GIRL:
        both_girls += 1
    if older == Kid.GIRL or younger == Kid.GIRL:
        either_girl += 1

print("P(both | older):", both_girls / older_girl)  
print("P(both | either): ", both_girls / either_girl)

P(both | older): 0.5007089325501317
P(both | either):  0.3311897106109325


PT_D = 0.99
PD = 0.0001
PT_negD = 0.01

numerator = PT_D * PD
denominator = numerator + PT_negD * (1 - PD)
PD_T = numerator/denominator

print(PD_T)

0.00980392156862745


99/(99 + 9999)

0.00980392156862745


from math import factorial

def P(r):
    return (factorial(10) * 2**(-10)) / (factorial(r) * factorial(10-r))

EX = sum([r * P(r) for r in range(11)])
print(EX)

5.0


sum(P(r) for r in range(11))

1.0


def uniform_pdf(x: float) -> float:
    return 1 if 0 <= x < 1 else 0


import matplotlib.pyplot as plt


xs = [x / 10.0 for x in range(-10, 20)]
plt.plot(xs,[uniform_pdf(x) for x in xs],'-',label='mu=0,sigma=1')
plt.show()


def uniform_cdf(x: float) -> float:
    if x < 0:   return 0
    elif x < 1: return x
    else:       return 1


plt.plot(xs,[uniform_cdf(x) for x in xs],'-',label='mu=0,sigma=1')
plt.show()


import math
SQRT_TWO_PI = math.sqrt(2 * math.pi)

def normal_pdf(x: float, mu: float = 0, sigma: float = 1) -> float:
    return (math.exp(-(x-mu) ** 2 / 2 / sigma ** 2) / (SQRT_TWO_PI * sigma))

import matplotlib.pyplot as 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=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()
plt.title("Various Normal pdfs")
plt.show()


def normal_cdf(x: float, mu: float = 0, sigma: float = 1) -> float:
    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.title("Various Normal cdfs")
plt.show()


P126 = normal_cdf(1.26)
P222 = normal_cdf(2.22)
print(f"P126: {P126}", 
     f"P222: {P222}", sep='\n')

P126: 0.8961653188786995
P222: 0.9867906161927438


def inverse_normal_cdf(p: float,
                       mu: float = 0,
                       sigma: float = 1,
                       tolerance: float = 0.00001) -> float:

    if mu != 0 or sigma != 1:
        return mu + sigma * inverse_normal_cdf(p, tolerance=tolerance)

    low_z = -10.0                      
    hi_z  =  10.0                      
    while hi_z - low_z > tolerance:
        mid_z = (low_z + hi_z) / 2     
        mid_p = normal_cdf(mid_z)      
        if mid_p < p:
            low_z = mid_z              
        else:
            hi_z = mid_z               

    return mid_z


inverse_normal_cdf(0.896165)

1.2600040435791016


import random

def bernoulli_trial(p: float) -> int:
    """Returns 1 with probability p and 0 with probability 1-p"""
    return 1 if random.random() < p else 0

def binomial(n: int, p: float) -> int:
    """Returns the sum of n bernoulli(p) trials"""
    return sum(bernoulli_trial(p) for _ in range(n))

from collections import Counter

def binomial_histogram(p: float, n: int, num_points: int) -> None:
    """Picks points from a Binomial(n, p) and plots their histogram"""
    data = [binomial(n, p) for _ in range(num_points)]

    # use a bar chart to show the actual binomial samples
    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
    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.title("Binomial Distribution vs. Normal Approximation")
    plt.show()


binomial_histogram(0.5,100,10000)


normal_cdf(-6)

9.865876449133282e-10

사건(r)	0	1	2	3	4	5	6	7	8	9	10
P(X=r)	P(X=0)	P(X=1)	P(X=2)	P(X=3)	P(X=4)	P(X=5)	P(X=6)	P(X=7)	P(X=8)	P(X=9)	P(X=10)

사건(x)	0	1	2
P(X=x)	1/4	1/2	1/4

사건(y)	1	2
P(Y=y)	2/3	1/3

사건(z)	1	2
P(Z=z)	1/2	1/2

확률¶

종속성과 독립성¶

독립사건 예제: 동전 두 번 던지기¶

종속사건 예제: 동전 두 번 던지기¶

조건부 확률¶

예제: 한 가족 내 두 아이들의 성별 맞추기¶

시뮬레이션(모의실험)¶

베이즈 정리¶

예제: 질병 여부 판단하기¶

확률변수와 확률분포¶

이산 확률분포¶

기댓값¶

예제¶

예제¶

예제¶

이산 확률분포와 조건부 확률¶

예제¶

연속 확률분포¶

균등 분포¶

균등분포 예제¶

확률밀도함수¶

정규분포¶

정규분포의 확률밀도함수¶

정규분포의 누적함수¶

표준정규분포¶

표준화¶

예제¶

예제¶

누적분포함수의 역함수¶

중심극한정리¶

이항분포와 중심극한정리¶

예제¶

사건(x)	0	1
P(X=x)	0.5	0.5

사건(y)	-100	100
P(Y=y)	0.5	0.5