الاحتمالات والإحصاء

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

class="tok-comment"># ── Normal distribution ───────────────────────────────────────────────────────
mu, sigma = class="tok-num">170, class="tok-num">10          class="tok-comment"># heights in cm
dist = stats.norm(mu, sigma)

x = np.linspace(class="tok-num">135, class="tok-num">205, class="tok-num">500)
pdf = dist.pdf(x)

class="tok-comment"># Probabilities
p_tall = class="tok-num">1 - dist.cdf(class="tok-num">190)           class="tok-comment"># P(X > class="tok-num">190)
p_range = dist.cdf(class="tok-num">180) - dist.cdf(class="tok-num">160)  class="tok-comment"># P(class="tok-num">160 < X < class="tok-num">180)
print(fclass="tok-str">"P(height > 190cm) = {p_tall:.4f}")
print(fclass="tok-str">"P(class="tok-num">160 < height < class="tok-num">180) = {p_range:.4f}")

class="tok-comment"># class="tok-num">68-class="tok-num">95-class="tok-num">99.7 rule
for k in [class="tok-num">1, class="tok-num">2, class="tok-num">3]:
    p = dist.cdf(mu + k*sigma) - dist.cdf(mu - k*sigma)
    print(fclass="tok-str">"P(μ ± {k}σ) = {p:.4f}")  class="tok-comment"># ≈ class="tok-num">0.68, class="tok-num">0.95, class="tok-num">0.997

class="tok-comment"># ── MLE — fitting a Normal distribution ──────────────────────────────────────
data = np.random.normal(class="tok-num">170, class="tok-num">10, size=class="tok-num">100)
mu_mle, sigma_mle = data.mean(), data.std()
print(fclass="tok-str">"\nMLE fit: μ̂={mu_mle:.2f}, σ̂={sigma_mle:.2f}")

class="tok-comment"># SciPy MLE (same result, handles any distribution)
mu_fit, sigma_fit = stats.norm.fit(data)
print(fclass="tok-str">"scipy fit: μ={mu_fit:.2f}, σ={sigma_fit:.2f}")

class="tok-comment"># ── Bayes' theorem ────────────────────────────────────────────────────────────
class="tok-comment"># Disease testing: prevalence class="tok-num">1%, test sensitivity class="tok-num">99%, specificity class="tok-num">95%
p_disease = class="tok-num">0.01
p_pos_given_disease = class="tok-num">0.99     class="tok-comment"># sensitivity
p_neg_given_healthy = class="tok-num">0.95    class="tok-comment"># specificity → P(pos|healthy) = class="tok-num">0.05

p_healthy = class="tok-num">1 - p_disease
p_pos_given_healthy = class="tok-num">1 - p_neg_given_healthy

class="tok-comment"># P(positive) = P(pos|disease)*P(disease) + P(pos|healthy)*P(healthy)
p_pos = p_pos_given_disease * p_disease + p_pos_given_healthy * p_healthy

class="tok-comment"># Bayes: P(disease | positive test)
p_disease_given_pos = (p_pos_given_disease * p_disease) / p_pos
print(fclass="tok-str">"\nP(disease | positive test) = {p_disease_given_pos:.4f}")  class="tok-comment"># ~class="tok-num">16.4%!
class="tok-comment"># Counterintuitive: despite class="tok-num">99% accurate test, only class="tok-num">16% chance with +ve result
class="tok-comment"># due to low base rate (prior) — base rate fallacy

class="tok-comment"># ── Hypothesis testing ────────────────────────────────────────────────────────
class="tok-comment"># Are two group means different?
group_a = np.random.normal(class="tok-num">5.0, class="tok-num">1.5, class="tok-num">50)
group_b = np.random.normal(class="tok-num">5.5, class="tok-num">1.5, class="tok-num">50)

t_stat, p_value = stats.ttest_ind(group_a, group_b)
print(fclass="tok-str">"\nt-test: t={t_stat:.3f}, p={p_value:.4f}")
print(class="tok-str">"Significant at α=class="tok-num">0.05:", p_value < class="tok-num">0.05)

class="tok-comment"># Bootstrap confidence interval for mean (distribution-free)
np.random.seed(class="tok-num">42)
boot_means = [np.random.choice(group_a, size=len(group_a), replace=True).mean()
              for _ in range(class="tok-num">10000)]
ci_low, ci_high = np.percentile(boot_means, [class="tok-num">2.5, class="tok-num">97.5])
print(fclass="tok-str">"class="tok-num">95% CI for group A mean: [{ci_low:.3f}, {ci_high:.3f}]")