import numpy as np
import matplotlib.pyplot as plt
from scipy import stats

# Generate normal data
data = np.random.normal(loc=100, scale=15, size=1000)
# loc = mean, scale = std dev

plt.hist(data, bins=30, density=True, alpha=0.7)
plt.title('Normal Distribution')
plt.show()

# 68-95-99.7 Rule
mean = 100
std = 15

# 68% of data within 1 std
print(f"68%: {mean - std} to {mean + std}")  # 85 to 115

# 95% of data within 2 std
print(f"95%: {mean - 2*std} to {mean + 2*std}")  # 70 to 130

# 99.7% of data within 3 std
print(f"99.7%: {mean - 3*std} to {mean + 3*std}")  # 55 to 145

from scipy.stats import norm

# IQ scores: mean=100, std=15

# P(IQ < 115)
p = norm.cdf(115, loc=100, scale=15)
print(f"P(IQ < 115): {p:.2%}")  # 84.13%

# P(IQ > 130)
p = 1 - norm.cdf(130, loc=100, scale=15)
print(f"P(IQ > 130): {p:.2%}")  # 2.28%

# P(85 < IQ < 115)
p = norm.cdf(115, 100, 15) - norm.cdf(85, 100, 15)
print(f"P(85 < IQ < 115): {p:.2%}")  # 68.27%

# Z-score formula: z = (x - mean) / std

x = 115
mean = 100
std = 15

z = (x - mean) / std
print(f"Z-score: {z}")  # 1.0

# Find percentile from z-score
percentile = norm.cdf(z)
print(f"Percentile: {percentile:.2%}")  # 84.13%

from scipy.stats import binom

# Flip coin 10 times, P(heads) = 0.5
# What's P(exactly 7 heads)?

n = 10  # trials
p = 0.5  # probability of success

# P(X = 7)
prob = binom.pmf(7, n, p)
print(f"P(7 heads): {prob:.2%}")  # 11.72%

# P(X >= 7)
prob = 1 - binom.cdf(6, n, p)
print(f"P(7+ heads): {prob:.2%}")  # 17.19%

from scipy.stats import poisson

# Average 3 customers per hour
# What's P(5 customers in an hour)?

lambda_rate = 3  # average rate

# P(X = 5)
prob = poisson.pmf(5, lambda_rate)
print(f"P(5 customers): {prob:.2%}")  # 10.08%

# P(X >= 5)
prob = 1 - poisson.cdf(4, lambda_rate)
print(f"P(5+ customers): {prob:.2%}")  # 18.47%

from scipy.stats import uniform

# Random number between 0 and 10
# P(value between 3 and 7)?

a = 0   # min
b = 10  # max

prob = uniform.cdf(7, a, b-a) - uniform.cdf(3, a, b-a)
print(f"P(3 < X < 7): {prob:.2%}")  # 40%

from scipy.stats import expon

# Average 1 customer every 20 minutes
# P(next customer in < 10 min)?

scale = 20  # average time

prob = expon.cdf(10, scale=scale)
print(f"P(< 10 min): {prob:.2%}")  # 39.35%

import numpy as np
from scipy import stats

# Exam scores: normally distributed
# Mean = 75, Std = 10

mean = 75
std = 10

# What score is needed for top 10%?
top_10_score = stats.norm.ppf(0.90, mean, std)
print(f"Top 10% score: {top_10_score:.1f}")  # 87.8

# What percent scored between 65 and 85?
percent = stats.norm.cdf(85, mean, std) - stats.norm.cdf(65, mean, std)
print(f"Between 65-85: {percent:.1%}")  # 68.3%

# Generate sample and verify
np.random.seed(42)
scores = np.random.normal(mean, std, 1000)

actual_percent = np.sum((scores >= 65) & (scores <= 85)) / len(scores)
print(f"Actual sample: {actual_percent:.1%}")