Why we have normal distribution formular

import numpy as np
import matplotlib.pyplot as plt

# Set random seed for consistent simulated coordinates
np.random.seed(42)

# Bivariate standard normal distribution parameter (spread scale sigma = 1)
sigma = 1.0

# Simulate 500 random dart throws based on independent normal errors
num_darts = 500
dart_x = np.random.normal(0, sigma, num_darts)
dart_y = np.random.normal(0, sigma, num_darts)

# Set up a single plot layout
fig, ax1 = plt.subplots(figsize=(6, 6))

# Plot: Simulated Dart Throws
ax1.scatter(dart_x, dart_y, color='red', alpha=0.6, edgecolors='k', s=20, label='Dart Landings')

# Draw concentric rings representing equal distance (r) from the center
angles = np.linspace(0, 2 * np.pi, 100)
for r in [1, 2, 3]:
    ax1.plot(r * np.cos(angles), r * np.sin(angles), linestyle='--', color='gray', alpha=0.7)

# Center crosshairs
ax1.axhline(0, color='black', linewidth=0.5)
ax1.axvline(0, color='black', linewidth=0.5)

# Axis configuration
ax1.set_xlim(-3, 3)
ax1.set_ylim(-3, 3)
ax1.set_aspect('equal')
ax1.set_title('Simulated Dart Throws')
ax1.set_xlabel('x error')
ax1.set_ylabel('y error')
ax1.legend()

plt.tight_layout()
plt.savefig('dartboard_visualization_single.png')

1 Reference

1. https://mccuan.math.gatech.edu/courses/3215/NormalDistributionSeanPeng.pdf