Gaussian Probability Density

import matplotlib.pyplot as plt
import numpy as np
from matplotlib.colors import LinearSegmentedColormap

# Create bivariate Gaussian
x = np.linspace(-4, 4, 200)
y = np.linspace(-4, 4, 200)
X, Y = np.meshgrid(x, y)

# Bivariate normal with correlation
mu_x, mu_y = 0, 0
sigma_x, sigma_y = 1, 1.5
rho = 0.6

z1 = (X - mu_x) / sigma_x
z2 = (Y - mu_y) / sigma_y
Z = np.exp(-1/(2*(1-rho**2)) * (z1**2 - 2*rho*z1*z2 + z2**2))
Z = Z / (2 * np.pi * sigma_x * sigma_y * np.sqrt(1 - rho**2))

# Modern dark theme
plt.style.use('dark_background')
fig, ax = plt.subplots(figsize=(10, 8))
fig.patch.set_facecolor('#0a0a0f')
ax.set_facecolor('#0a0a0f')

# Custom colormap - vibrant purple-pink gradient
colors = ['#0a0a0f', '#1a0a2e', '#2d1a4a', '#4a2a6a', '#6a3a8a', 
          '#8a4aaa', '#aa5aca', '#ca6aea', '#ea7aff', '#ff8aff']
cmap = LinearSegmentedColormap.from_list('neon_purple', colors, N=256)

# Filled contour with glow effect
cf = ax.contourf(X, Y, Z, levels=25, cmap=cmap)

# Subtle contour lines
cs = ax.contour(X, Y, Z, levels=12, colors='#ff8aff', linewidths=0.4, alpha=0.3)

# Mark center with glow
ax.scatter([mu_x], [mu_y], color='#ff8aff', s=150, marker='+', linewidths=3, zorder=5)
ax.scatter([mu_x], [mu_y], color='#ff8aff', s=300, marker='+', linewidths=1, alpha=0.3, zorder=4)

# Styled colorbar
cbar = plt.colorbar(cf, ax=ax, pad=0.02, shrink=0.85)
cbar.ax.set_facecolor('#0a0a0f')
cbar.set_label('Probability Density', fontsize=11, color='#e0e0ff', labelpad=10)
cbar.ax.yaxis.set_tick_params(color='#e0e0ff')
cbar.outline.set_edgecolor('#3a3a5a')
plt.setp(plt.getp(cbar.ax.axes, 'yticklabels'), color='#e0e0ff', fontsize=9)

# Labels with modern styling
ax.set_xlabel('X', fontsize=13, color='#e0e0ff', fontweight='600', labelpad=10)
ax.set_ylabel('Y', fontsize=13, color='#e0e0ff', fontweight='600', labelpad=10)
ax.set_title('Bivariate Gaussian Distribution', fontsize=16, color='white', 
             fontweight='bold', pad=20)

# Style axes
ax.tick_params(colors='#a0a0d0', labelsize=10, length=0)
for spine in ax.spines.values():
    spine.set_visible(False)

# Add subtle grid
ax.grid(True, alpha=0.1, color='#ffffff', linestyle='-', linewidth=0.5)

ax.set_aspect('equal')
plt.tight_layout()
plt.show()