Gaussian approximation

 

Gaussian Approximation – Explained with Diagrams 📊

Gaussian Approximation refers to the concept that, under certain conditions, a distribution can be well approximated by a Gaussian (Normal) distribution. This is widely used in statistics, probability, and machine learning.

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1️⃣ Why Gaussian Approximation Matters?

• Many real-world phenomena follow a normal distribution (e.g., height, test scores).

• Even non-normal distributions can often be approximated by a Gaussian under the Central Limit Theorem (CLT).

• Many statistical and machine learning models assume normality for inference.

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2️⃣ Key Examples of Gaussian Approximation

We'll visualize:

1. Binomial to Normal Approximation (for large trials).

2. Poisson to Normal Approximation (for large mean).

3. Skewed Data Becoming Normal via CLT.

Let's generate diagrams to illustrate these cases! 🎨📊

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📜 Python Code for Visualizing Gaussian Approximation

We will:

• Plot a Binomial and Poisson distribution.

• Overlay a Normal (Gaussian) curve to see the approximation.

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

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

### 1️⃣ Binomial Distribution Approaching Normal ###
n, p = 50, 0.5  # Large number of trials
binom_x = np.arange(0, n+1)
binom_pmf = stats.binom.pmf(binom_x, n, p)

# Normal Approximation
normal_x = np.linspace(0, n, 100)
normal_pdf = stats.norm.pdf(normal_x, loc=n*p, scale=np.sqrt(n*p*(1-p)))

# Plotting Binomial vs Normal
axes[0].bar(binom_x, binom_pmf, color='blue', alpha=0.6, label="Binomial PMF")
axes[0].plot(normal_x, normal_pdf, color='red', linewidth=2, label="Normal Approximation")
axes[0].set_title("Binomial to Gaussian Approximation (n=50)")
axes[0].legend()

### 2️⃣ Poisson Distribution Approaching Normal ###
lambda_val = 30  # Large mean
poisson_x = np.arange(0, lambda_val*2)
poisson_pmf = stats.poisson.pmf(poisson_x, lambda_val)

# Normal Approximation
normal_pdf_p = stats.norm.pdf(poisson_x, loc=lambda_val, scale=np.sqrt(lambda_val))

# Plotting Poisson vs Normal
axes[1].bar(poisson_x, poisson_pmf, color='green', alpha=0.6, label="Poisson PMF")
axes[1].plot(poisson_x, normal_pdf_p, color='red', linewidth=2, label="Normal Approximation")
axes[1].set_title("Poisson to Gaussian Approximation (λ=30)")
axes[1].legend()

plt.show()

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📊 Expected Diagrams & Explanation

1. Left Plot (Binomial to Normal)

   • The blue bars show a Binomial distribution (n=50, p=0.5).

   • The red curve is the Gaussian approximation.

   • For large $n$, the binomial distribution becomes bell-shaped and closely follows the normal distribution.

2. Right Plot (Poisson to Normal)

   • The green bars represent a Poisson distribution ($\lambda = 30$).

   • The red curve is the corresponding normal approximation.

   • For large $\lambda$, Poisson becomes symmetric and bell-shaped, making it well-approximated by a normal distribution.

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📌 Key Takeaways

✅ The Binomial distribution is approximately normal when $n$ is large and $p$ is not too close to 0 or 1.
✅ The Poisson distribution becomes normal when $\lambda$ is large.
✅ Many real-world problems can be modeled using normal distributions, simplifying analysis!

Here is the visualization of Gaussian Approximation in action! 📊

1️⃣ Left Plot (Binomial to Normal):

• The blue bars show a Binomial distribution ($n=50, p=0.5$).

• The red curve represents the Normal Approximation.

• As $n$ increases, the binomial distribution closely follows a normal distribution.

2️⃣ Right Plot (Poisson to Normal):

• The green bars show a Poisson distribution ($\lambda = 30$).

• The red curve is the Normal Approximation.

• For large $\lambda$, Poisson distribution becomes more symmetric and resembles a normal distribution.

This confirms that many discrete distributions can be well approximated by a Gaussian, making statistical analysis much simpler! 🚀