NumPy: مشروع — محاكاة مونت كارلو

import numpy as np

rng = np.random.default_rng(42)

N = 100_000
x = rng.random(N)
y = rng.random(N)
inside = (x**2 + y**2) <= 1.0
pi_est = 4 * inside.mean()

print(f"Pi estimate: {pi_est:.6f}")
print(f"Actual pi:   {np.pi:.6f}")
print(f"Error:       {abs(pi_est - np.pi):.6f}")

print("=== Monte Carlo Estimation of Pi ===\n")

for N in [1_000, 10_000, 100_000, 1_000_000]:
    x = rng.random(N)
    y = rng.random(N)
    inside = (x**2 + y**2) <= 1.0
    pi_est = 4 * inside.mean()
    error = abs(pi_est - np.pi)
    print(f"N={N:>9,d}  Pi≈{pi_est:.6f}  Error={error:.6f}")

N = 1000
n_sim = 10_000

steps = rng.choice([-1, 1], size=(n_sim, N))
final_pos = steps.sum(axis=1)

print(f"Final position statistics ({n_sim} simulations):")
print(f"  Mean: {final_pos.mean():.2f}  (theory: 0)")
print(f"  Std:  {final_pos.std():.2f}  (theory: {np.sqrt(N):.2f})")
print(f"  Min:  {final_pos.min()}")
print(f"  Max:  {final_pos.max()}")

S0 = 100
K = 105
r = 0.05
sigma = 0.20
T = 1.0

for n_sim in [10_000, 100_000, 1_000_000]:
    z = rng.standard_normal(n_sim)
    ST = S0 * np.exp((r - 0.5 * sigma**2) * T + sigma * np.sqrt(T) * z)

    call_payoff = np.maximum(ST - K, 0)
    put_payoff = np.maximum(K - ST, 0)

    call_price = np.exp(-r * T) * call_payoff.mean()
    put_price = np.exp(-r * T) * put_payoff.mean()
    call_se = np.exp(-r * T) * call_payoff.std() / np.sqrt(n_sim)

    print(f"N={n_sim:>9,d}")
    print(f"  Call: {call_price:.4f} ± {1.96*call_se:.4f}")
    print(f"  Put:  {put_price:.4f}")

data = rng.exponential(scale=2.0, size=50)
n = len(data)
B = 10_000

idx = rng.integers(0, n, size=(B, n))
boot_samples = data[idx]
boot_means = boot_samples.mean(axis=1)

ci = np.percentile(boot_means, [2.5, 97.5])
print(f"Bootstrap 95% CI for mean: [{ci[0]:.3f}, {ci[1]:.3f}]")
print(f"Sample mean: {data.mean():.3f}")

import numpy as np

rng = np.random.default_rng(42)
N = 100_000
x = rng.random(N)
y = rng.random(N)
inside = (x**2 + y**2) <= 1.0
pi_est = 4 * inside.mean()
print(f"Estimated Pi: {pi_est:.6f}")
print(f"Actual Pi:    {np.pi:.6f}")
print(f"Error:        {abs(pi_est - np.pi):.6f}")

# تم التنفيذ بنجاح

import numpy as np

rng = np.random.default_rng(42)
n_steps = 1000
n_simulations = 100

# جميع المسارات مرة واحدة باستخدام التوجيه
steps = rng.choice([-1, 1], size=(n_simulations, n_steps))
positions = np.cumsum(steps, axis=1)

final = positions[:, -1]
print(f"Mean final position: {final.mean():.2f}")
print(f"Std final position:  {final.std():.2f}")
print(f"Min: {final.min()}, Max: {final.max()}")

# تم التنفيذ بنجاح

import numpy as np

rng = np.random.default_rng(42)
data = rng.exponential(scale=2.0, size=50)

n = len(data)
B = 10_000
indices = rng.integers(0, n, size=(B, n))
boot_means = data[indices].mean(axis=1)

ci = np.percentile(boot_means, [2.5, 97.5])
print(f"Sample mean: {data.mean():.3f}")
print(f"95% CI: [{ci[0]:.3f}, {ci[1]:.3f}]")
print(f"CI width: {ci[1] - ci[0]:.3f}")

# تم التنفيذ بنجاح