# ============= CENTRAL LIMIT THEOREM =============

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
from scipy import stats

print("="*60)
print("CENTRAL LIMIT THEOREM DEMONSTRATION")
print("="*60)

# Create a highly skewed population (lognormal)
np.random.seed(42)
population = np.random.lognormal(2, 1, 10000)

print(f"\n📊 Population Statistics:")
print(f"  Mean: {population.mean():.2f}")
print(f"  Median: {np.median(population):.2f}")
print(f"  Skewness: {pd.Series(population).skew():.3f}")

# Draw samples of different sizes
sample_sizes = [5, 10, 30, 100]
n_samples = 1000

fig, axes = plt.subplots(2, 2, figsize=(14, 10))

for i, sample_size in enumerate(sample_sizes):
    row = i // 2
    col = i % 2
    
    # Draw multiple samples and calculate means
    sample_means = []
    for _ in range(n_samples):
        sample = np.random.choice(population, size=sample_size, replace=True)
        sample_means.append(sample.mean())
    
    # Plot distribution of sample means
    ax = axes[row, col]
    ax.hist(sample_means, bins=30, edgecolor='black', alpha=0.7, density=True)
    
    # Overlay normal distribution
    mu = np.mean(sample_means)
    sigma = np.std(sample_means)
    x = np.linspace(mu - 4*sigma, mu + 4*sigma, 100)
    y = stats.norm.pdf(x, mu, sigma)
    ax.plot(x, y, 'r-', linewidth=2, label='Normal Fit')
    
    # Add vertical line for population mean
    ax.axvline(population.mean(), color='black', linestyle='--', label=f'Population Mean: {population.mean():.2f}')
    
    ax.set_title(f'Sample Size: {sample_size}', fontsize=12)
    ax.set_xlabel('Sample Mean')
    ax.set_ylabel('Density')
    ax.legend()
    ax.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('central_limit_theorem.png', dpi=300)
plt.show()

print("\n💡 Key Insight:")
print("  • As sample size increases, distribution of sample means becomes more normal")
print("  • Mean of sample means approaches population mean")
print("  • Standard deviation of sample means decreases (Standard Error)")

# ============= STANDARD ERROR AND CONFIDENCE INTERVALS =============

def calculate_confidence_interval(data, confidence=0.95):
    """Calculate confidence interval for the mean."""
    n = len(data)
    mean = np.mean(data)
    std = np.std(data, ddof=1)
    se = std / np.sqrt(n)
    
    # t-value for confidence level
    t_value = stats.t.ppf((1 + confidence) / 2, n - 1)
    
    margin = t_value * se
    ci_lower = mean - margin
    ci_upper = mean + margin
    
    return {
        'mean': mean,
        'std': std,
        'se': se,
        'ci_lower': ci_lower,
        'ci_upper': ci_upper,
        'margin': margin
    }

# Apply to banking data
print("\n" + "="*60)
print("CONFIDENCE INTERVALS FOR BANKING METRICS")
print("="*60)

# Create sample data
np.random.seed(42)
n = 100

# Monthly transaction amounts for different customer segments
premium_transactions = np.random.normal(350, 80, n)
standard_transactions = np.random.normal(200, 60, n)
basic_transactions = np.random.normal(120, 40, n)

print("\n📊 Monthly Transaction Amounts:")
print(f"  Premium: Mean={premium_transactions.mean():.2f}, Std={premium_transactions.std():.2f}")
print(f"  Standard: Mean={standard_transactions.mean():.2f}, Std={standard_transactions.std():.2f}")
print(f"  Basic: Mean={basic_transactions.mean():.2f}, Std={basic_transactions.std():.2f}")

# Calculate confidence intervals
for segment, data in [('Premium', premium_transactions), 
                      ('Standard', standard_transactions), 
                      ('Basic', basic_transactions)]:
    ci = calculate_confidence_interval(data, confidence=0.95)
    print(f"\n{segment} Segment (95% CI):")
    print(f"  Mean: ${ci['mean']:.2f}")
    print(f"  Standard Error: ${ci['se']:.2f}")
    print(f"  CI: [${ci['ci_lower']:.2f}, ${ci['ci_upper']:.2f}]")
    print(f"  Margin of Error: ±${ci['margin']:.2f}")

# Visualize confidence intervals
fig, ax = plt.subplots(figsize=(10, 6))

segments = ['Premium', 'Standard', 'Basic']
means = [np.mean(premium_transactions), np.mean(standard_transactions), np.mean(basic_transactions)]
cis = [calculate_confidence_interval(data, 0.95) for data in [premium_transactions, standard_transactions, basic_transactions]]

for i, (segment, mean, ci) in enumerate(zip(segments, means, cis)):
    ax.errorbar(i, mean, yerr=ci['margin'], fmt='o', capsize=5, capthick=2, 
                markersize=10, color='blue', ecolor='red', elinewidth=2)
    ax.text(i, mean + ci['margin'] + 5, f'${mean:.0f}', ha='center', va='bottom')

ax.set_xticks(range(len(segments)))
ax.set_xticklabels(segments)
ax.set_ylabel('Monthly Transaction Amount ($)')
ax.set_title('95% Confidence Intervals by Customer Segment', fontsize=14)
ax.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('confidence_intervals.png', dpi=300)
plt.show()

# ============= ONE-SAMPLE T-TEST =============

print("\n" + "="*60)
print("ONE-SAMPLE T-TEST")
print("="*60)

# Example: Is the average credit score different from 700?
credit_scores = np.random.normal(685, 50, 100)
known_mean = 700

# Perform one-sample t-test
t_stat, p_value = stats.ttest_1samp(credit_scores, known_mean)

print(f"\n📊 Credit Score Analysis:")
print(f"  Sample Mean: {credit_scores.mean():.2f}")
print(f"  Hypothesized Mean: {known_mean}")
print(f"  Sample Std: {credit_scores.std():.2f}")
print(f"  Sample Size: {len(credit_scores)}")
print(f"  T-Statistic: {t_stat:.3f}")
print(f"  P-Value: {p_value:.4f}")

if p_value < 0.05:
    print(f"  Result: REJECT null hypothesis (p < 0.05)")
    print(f"  Interpretation: The average credit score IS significantly different from {known_mean}")
else:
    print(f"  Result: FAIL TO REJECT null hypothesis (p >= 0.05)")
    print(f"  Interpretation: No significant difference from {known_mean}")

# Visualize
fig, axes = plt.subplots(1, 2, figsize=(14, 5))

# Histogram with hypothesized mean
ax = axes[0]
credit_scores.hist(bins=20, ax=ax, edgecolor='black', alpha=0.7)
ax.axvline(credit_scores.mean(), color='red', linestyle='--', linewidth=2, label=f'Sample Mean: {credit_scores.mean():.2f}')
ax.axvline(known_mean, color='blue', linestyle='--', linewidth=2, label=f'Hypothesized Mean: {known_mean}')
ax.set_title('Credit Score Distribution', fontsize=12)
ax.set_xlabel('Credit Score')
ax.set_ylabel('Frequency')
ax.legend()
ax.grid(True, alpha=0.3)

# T-distribution
ax = axes[1]
x = np.linspace(-4, 4, 1000)
y = stats.t.pdf(x, len(credit_scores) - 1)
ax.plot(x, y, 'b-', linewidth=2)
ax.axvline(t_stat, color='red', linestyle='--', linewidth=2, label=f'T-statistic: {t_stat:.3f}')
ax.axvline(-t_stat, color='red', linestyle='--', linewidth=2)

# Shade critical regions
alpha = 0.05
t_critical = stats.t.ppf(1 - alpha/2, len(credit_scores) - 1)
ax.fill_between(x, 0, y, where=(x > t_critical), color='red', alpha=0.3, label='Rejection Region')
ax.fill_between(x, 0, y, where=(x < -t_critical), color='red', alpha=0.3)

ax.set_title('T-Distribution', fontsize=12)
ax.set_xlabel('T-Statistic')
ax.set_ylabel('Density')
ax.legend()
ax.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('one_sample_ttest.png', dpi=300)
plt.show()

# ============= TWO-SAMPLE T-TEST =============

print("\n" + "="*60)
print("TWO-SAMPLE T-TEST")
print("="*60)

# Example: Compare transaction amounts between Premium and Standard customers
premium_transactions = np.random.normal(350, 80, 100)
standard_transactions = np.random.normal(200, 60, 100)

# Perform two-sample t-test
t_stat, p_value = stats.ttest_ind(premium_transactions, standard_transactions)

print(f"\n📊 Transaction Amount Comparison:")
print(f"  Premium: Mean={premium_transactions.mean():.2f}, Std={premium_transactions.std():.2f}")
print(f"  Standard: Mean={standard_transactions.mean():.2f}, Std={standard_transactions.std():.2f}")
print(f"  Difference: {premium_transactions.mean() - standard_transactions.mean():.2f}")
print(f"  T-Statistic: {t_stat:.3f}")
print(f"  P-Value: {p_value:.4f}")

if p_value < 0.05:
    print(f"  Result: REJECT null hypothesis")
    print(f"  Interpretation: Premium customers have significantly different transaction amounts")
else:
    print(f"  Result: FAIL TO REJECT null hypothesis")
    print(f"  Interpretation: No significant difference in transaction amounts")

# Visualize
fig, axes = plt.subplots(1, 2, figsize=(14, 5))

# Box plots
ax = axes[0]
data = [premium_transactions, standard_transactions]
ax.boxplot(data, labels=['Premium', 'Standard'])
ax.set_title('Transaction Amounts by Segment', fontsize=12)
ax.set_ylabel('Amount ($)')
ax.grid(True, alpha=0.3)

# Histograms
ax = axes[1]
ax.hist(premium_transactions, bins=20, alpha=0.5, label='Premium', edgecolor='black')
ax.hist(standard_transactions, bins=20, alpha=0.5, label='Standard', edgecolor='black')
ax.axvline(premium_transactions.mean(), color='blue', linestyle='--', label=f'Premium Mean: {premium_transactions.mean():.2f}')
ax.axvline(standard_transactions.mean(), color='orange', linestyle='--', label=f'Standard Mean: {standard_transactions.mean():.2f}')
ax.set_title('Transaction Amount Distributions', fontsize=12)
ax.set_xlabel('Amount ($)')
ax.set_ylabel('Frequency')
ax.legend()
ax.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('two_sample_ttest.png', dpi=300)
plt.show()

# ============= PAIRED T-TEST =============

print("\n" + "="*60)
print("PAIRED T-TEST")
print("="*60)

# Example: Impact of financial education on credit scores
np.random.seed(42)
n_customers = 50

# Before education
before_scores = np.random.normal(640, 50, n_customers)

# After education (some improvement)
improvement = np.random.normal(20, 15, n_customers)
after_scores = before_scores + improvement

# Perform paired t-test
t_stat, p_value = stats.ttest_rel(before_scores, after_scores)

print(f"\n📊 Financial Education Impact:")
print(f"  Before: Mean={before_scores.mean():.2f}, Std={before_scores.std():.2f}")
print(f"  After: Mean={after_scores.mean():.2f}, Std={after_scores.std():.2f}")
print(f"  Average Improvement: {after_scores.mean() - before_scores.mean():.2f}")
print(f"  T-Statistic: {t_stat:.3f}")
print(f"  P-Value: {p_value:.4f}")

if p_value < 0.05:
    print(f"  Result: REJECT null hypothesis")
    print(f"  Interpretation: Financial education significantly improved credit scores")
else:
    print(f"  Result: FAIL TO REJECT null hypothesis")
    print(f"  Interpretation: No significant improvement detected")

# Visualize
fig, axes = plt.subplots(1, 2, figsize=(14, 5))

# Before and after distributions
ax = axes[0]
ax.hist(before_scores, bins=15, alpha=0.5, label='Before', edgecolor='black')
ax.hist(after_scores, bins=15, alpha=0.5, label='After', edgecolor='black')
ax.axvline(before_scores.mean(), color='blue', linestyle='--', label=f'Before Mean: {before_scores.mean():.2f}')
ax.axvline(after_scores.mean(), color='orange', linestyle='--', label=f'After Mean: {after_scores.mean():.2f}')
ax.set_title('Credit Score Distribution - Before vs After', fontsize=12)
ax.set_xlabel('Credit Score')
ax.set_ylabel('Frequency')
ax.legend()
ax.grid(True, alpha=0.3)

# Paired differences
ax = axes[1]
differences = after_scores - before_scores
ax.hist(differences, bins=15, edgecolor='black', alpha=0.7)
ax.axvline(0, color='red', linestyle='--', linewidth=2, label='No Change')
ax.axvline(differences.mean(), color='blue', linestyle='--', linewidth=2, label=f'Mean Improvement: {differences.mean():.2f}')
ax.set_title('Distribution of Improvements', fontsize=12)
ax.set_xlabel('Score Improvement')
ax.set_ylabel('Frequency')
ax.legend()
ax.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('paired_ttest.png', dpi=300)
plt.show()

# ============= ANALYSIS OF VARIANCE (ANOVA) =============

print("\n" + "="*60)
print("ANALYSIS OF VARIANCE (ANOVA)")
print("="*60)

# Example: Compare transaction amounts across all three segments
np.random.seed(42)
premium = np.random.normal(350, 80, 100)
standard = np.random.normal(200, 60, 100)
basic = np.random.normal(120, 40, 100)

# Perform one-way ANOVA
f_stat, p_value = stats.f_oneway(premium, standard, basic)

print(f"\n📊 Transaction Amount by Segment:")
print(f"  Premium: Mean={premium.mean():.2f}, Std={premium.std():.2f}")
print(f"  Standard: Mean={standard.mean():.2f}, Std={standard.std():.2f}")
print(f"  Basic: Mean={basic.mean():.2f}, Std={basic.std():.2f}")
print(f"\n  F-Statistic: {f_stat:.3f}")
print(f"  P-Value: {p_value:.4f}")

if p_value < 0.05:
    print(f"  Result: REJECT null hypothesis")
    print(f"  Interpretation: There is a significant difference between at least two segments")
    
    # Post-hoc analysis (Tukey's HSD)
    from statsmodels.stats.multicomp import pairwise_tukeyhsd
    
    # Prepare data for post-hoc
    all_data = np.concatenate([premium, standard, basic])
    all_groups = np.concatenate([['Premium']*100, ['Standard']*100, ['Basic']*100])
    
    tukey = pairwise_tukeyhsd(all_data, all_groups, alpha=0.05)
    print("\n  Post-hoc Analysis (Tukey's HSD):")
    print(tukey)
    
else:
    print(f"  Result: FAIL TO REJECT null hypothesis")
    print(f"  Interpretation: No significant difference between segments")

# Visualize
fig, axes = plt.subplots(1, 2, figsize=(14, 5))

# Box plots
ax = axes[0]
data = [premium, standard, basic]
ax.boxplot(data, labels=['Premium', 'Standard', 'Basic'])
ax.set_title('Transaction Amounts by Segment', fontsize=12)
ax.set_ylabel('Amount ($)')
ax.grid(True, alpha=0.3)

# Violin plots
ax = axes[1]
sns.violinplot(data=[premium, standard, basic], ax=ax)
ax.set_xticklabels(['Premium', 'Standard', 'Basic'])
ax.set_title('Transaction Amount Distributions', fontsize=12)
ax.set_ylabel('Amount ($)')
ax.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('anova_analysis.png', dpi=300)
plt.show()

# ============= CHI-SQUARE TESTS =============

print("\n" + "="*60)
print("CHI-SQUARE TESTS")
print("="*60)

# Example: Is there an association between customer segment and loan default?
np.random.seed(42)

# Create contingency table
n_customers = 500
segments = np.random.choice(['Premium', 'Standard', 'Basic'], n_customers, p=[0.15, 0.55, 0.30])

# Simulate defaults based on segment
default_prob = {'Premium': 0.02, 'Standard': 0.05, 'Basic': 0.10}
defaults = np.array([1 if np.random.random() < default_prob[seg] else 0 for seg in segments])

# Create contingency table
contingency = pd.crosstab(segments, defaults, margins=False)

print("\n📊 Contingency Table - Segment vs Default:")
print(contingency)

# Perform chi-square test
from scipy.stats import chi2_contingency
chi2, p_value, dof, expected = chi2_contingency(contingency)

print(f"\n  Chi-Square Statistic: {chi2:.3f}")
print(f"  P-Value: {p_value:.4f}")
print(f"  Degrees of Freedom: {dof}")

if p_value < 0.05:
    print(f"  Result: REJECT null hypothesis")
    print(f"  Interpretation: There is a significant association between segment and default")
else:
    print(f"  Result: FAIL TO REJECT null hypothesis")
    print(f"  Interpretation: No significant association between segment and default")

# Visualize
fig, axes = plt.subplots(1, 2, figsize=(14, 5))

# Stacked bar chart
ax = axes[0]
contingency_pct = contingency.div(contingency.sum(axis=1), axis=0)
contingency_pct.plot(kind='bar', stacked=True, ax=ax)
ax.set_title('Default Rate by Segment', fontsize=12)
ax.set_xlabel('Segment')
ax.set_ylabel('Proportion')
ax.legend(['No Default', 'Default'], title='Status')
ax.grid(True, alpha=0.3)

# Heatmap of expected vs observed
ax = axes[1]
sns.heatmap(contingency, annot=True, fmt='d', cmap='Blues', ax=ax)
ax.set_title('Observed Frequencies', fontsize=12)
ax.set_xlabel('Default')
ax.set_ylabel('Segment')

plt.tight_layout()
plt.savefig('chi_square_analysis.png', dpi=300)
plt.show()

# ============= NON-PARAMETRIC TESTS =============

print("\n" + "="*60)
print("NON-PARAMETRIC TESTS")
print("="*60)

# Create skewed data (lognormal)
np.random.seed(42)
group_a = np.random.lognormal(3, 0.5, 100)
group_b = np.random.lognormal(3.5, 0.5, 100)

print("\n📊 Data with Non-Normal Distribution:")
print(f"  Group A: Mean={group_a.mean():.2f}, Median={np.median(group_a):.2f}, Skew={pd.Series(group_a).skew():.3f}")
print(f"  Group B: Mean={group_b.mean():.2f}, Median={np.median(group_b):.2f}, Skew={pd.Series(group_b).skew():.3f}")

# Mann-Whitney U test (non-parametric alternative to t-test)
u_stat, p_value = stats.mannwhitneyu(group_a, group_b, alternative='two-sided')

print(f"\n📊 Mann-Whitney U Test:")
print(f"  U-Statistic: {u_stat:.3f}")
print(f"  P-Value: {p_value:.4f}")

if p_value < 0.05:
    print(f"  Result: Significant difference between groups")
else:
    print(f"  Result: No significant difference between groups")

# Kruskal-Wallis test (non-parametric alternative to ANOVA)
# Add a third group
group_c = np.random.lognormal(4, 0.5, 100)
h_stat, p_value = stats.kruskal(group_a, group_b, group_c)

print(f"\n📊 Kruskal-Wallis Test:")
print(f"  H-Statistic: {h_stat:.3f}")
print(f"  P-Value: {p_value:.4f}")

if p_value < 0.05:
    print(f"  Result: Significant difference between at least two groups")
else:
    print(f"  Result: No significant difference between groups")

# Visualize
fig, axes = plt.subplots(1, 2, figsize=(14, 5))

# Box plots
ax = axes[0]
data = [group_a, group_b, group_c]
ax.boxplot(data, labels=['Group A', 'Group B', 'Group C'])
ax.set_title('Non-Normal Data Comparison', fontsize=12)
ax.set_ylabel('Value')
ax.grid(True, alpha=0.3)

# QQ plots for normality check
ax = axes[1]
stats.probplot(group_a, dist="norm", plot=ax)
ax.set_title('QQ Plot - Group A', fontsize=12)

plt.tight_layout()
plt.savefig('nonparametric_tests.png', dpi=300)
plt.show()

# ============= BUSINESS APPLICATIONS =============

print("\n" + "="*60)
print("BUSINESS APPLICATIONS OF STATISTICAL INFERENCE")
print("="*60)

applications = [
    {
        'application': 'Credit Scoring Model Validation',
        'test': 't-test',
        'use_case': 'Compare model predictions vs actual outcomes',
        'impact': 'Ensure model accuracy, comply with SR 11-7'
    },
    {
        'application': 'Marketing Campaign Effectiveness',
        'test': 'paired t-test',
        'use_case': 'Compare customer spending before and after campaign',
        'impact': 'Optimize marketing spend, increase ROI'
    },
    {
        'application': 'Customer Segment Analysis',
        'test': 'ANOVA',
        'use_case': 'Compare metrics across customer segments',
        'impact': 'Targeted product offerings, improve profitability'
    },
    {
        'application': 'Fraud Detection',
        'test': 'chi-square',
        'use_case': 'Test association between transaction characteristics and fraud',
        'impact': 'Improve fraud detection, reduce losses'
    },
    {
        'application': 'A/B Testing',
        'test': 'two-sample t-test',
        'use_case': 'Compare performance of different strategies',
        'impact': 'Data-driven decision making, optimize outcomes'
    }
]

for app in applications:
    print(f"\n📊 {app['application']}:")
    print(f"  Test: {app['test']}")
    print(f"  Use Case: {app['use_case']}")
    print(f"  Business Impact: {app['impact']}")

# ============= REGULATORY CONSIDERATIONS =============

print("\n" + "="*60)
print("REGULATORY CONSIDERATIONS")
print("="*60)

regulatory = {
    'SR 11-7': {
        'requirement': 'Model validation must include statistical testing',
        'implication': 'Document all hypothesis tests and their results'
    },
    'BASEL III': {
        'requirement': 'Risk models must be statistically sound',
        'implication': 'Use appropriate tests for model calibration'
    },
    'Fair Lending': {
        'requirement': 'No discrimination in lending decisions',
        'implication': 'Use statistical tests to check for bias'
    },
    'GDPR': {
        'requirement': 'Right to explanation of automated decisions',
        'implication': 'Statistical results must be interpretable and explainable'
    }
}

for reg, details in regulatory.items():
    print(f"\n📋 {reg}:")
    print(f"  Requirement: {details['requirement']}")
    print(f"  Implication: {details['implication']}")