Apprendre Fundamental Properties and Intuition of the Gaussian Distribution

The Gaussian distribution, also known as the normal distribution, is one of the most fundamental probability distributions in statistics and machine learning. It is characterized by its iconic bell-shaped curve, which is defined mathematically by two parameters: the mean (often denoted as $μ$ ) and the variance (denoted as $σ²$ ). The mean determines the center or location of the distribution, while the variance controls the spread or width of the curve.

f(x|\mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)

The mean ( $μ$ ): sets the peak of the bell curve. It is the average value around which the data is distributed;
The variance ( $σ²$ ): measures how much the data is spread out around the mean. A larger variance results in a wider, flatter curve; a smaller variance creates a narrower, taller curve.

This distribution is central to machine learning because many real-world phenomena tend to follow a Gaussian pattern due to the Central Limit Theorem, and it forms the foundation for many algorithms and statistical models.


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import numpy as np
import matplotlib.pyplot as plt

# Generate x values
x = np.linspace(-10, 10, 400)

# Define different means and variances
params = [
    {"mu": 0, "sigma2": 1, "label": "μ=0, σ²=1"},
    {"mu": 0, "sigma2": 4, "label": "μ=0, σ²=4"},
    {"mu": 2, "sigma2": 1, "label": "μ=2, σ²=1"},
    {"mu": -2, "sigma2": 0.5, "label": "μ=-2, σ²=0.5"},
]

plt.figure(figsize=(8, 5))
for p in params:
    mu = p["mu"]
    sigma = np.sqrt(p["sigma2"])
    y = (1 / (np.sqrt(2 * np.pi) * sigma)) * np.exp(-0.5 * ((x - mu) / sigma) ** 2)
    plt.plot(x, y, label=p["label"])

plt.title("Gaussian Distributions with Varying Means and Variances")
plt.xlabel("x")
plt.ylabel("Probability Density")
plt.legend()
plt.grid(True)
plt.show()

By examining the plotted curves, you can see how the mean and variance parameters shape the Gaussian distribution. Shifting the mean ( $μ$ ) moves the center of the curve left or right along the x-axis, without affecting its width. Adjusting the variance ( $σ²$ ) changes the spread: a larger variance results in a flatter, wider curve, while a smaller variance produces a taller, narrower peak. These properties are crucial in machine learning, where the mean can represent an expected value and the variance reflects uncertainty or variability in your data. Understanding how these parameters influence the distribution's shape helps you interpret model assumptions and the behavior of algorithms that rely on the Gaussian distribution.

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Section 2. Chapitre 1

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f(x|\mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)

The mean ( $μ$ ): sets the peak of the bell curve. It is the average value around which the data is distributed;
The variance ( $σ²$ ): measures how much the data is spread out around the mean. A larger variance results in a wider, flatter curve; a smaller variance creates a narrower, taller curve.


              123456789101112131415161718192021222324252627
            
import numpy as np
import matplotlib.pyplot as plt

# Generate x values
x = np.linspace(-10, 10, 400)

# Define different means and variances
params = [
    {"mu": 0, "sigma2": 1, "label": "μ=0, σ²=1"},
    {"mu": 0, "sigma2": 4, "label": "μ=0, σ²=4"},
    {"mu": 2, "sigma2": 1, "label": "μ=2, σ²=1"},
    {"mu": -2, "sigma2": 0.5, "label": "μ=-2, σ²=0.5"},
]

plt.figure(figsize=(8, 5))
for p in params:
    mu = p["mu"]
    sigma = np.sqrt(p["sigma2"])
    y = (1 / (np.sqrt(2 * np.pi) * sigma)) * np.exp(-0.5 * ((x - mu) / sigma) ** 2)
    plt.plot(x, y, label=p["label"])

plt.title("Gaussian Distributions with Varying Means and Variances")
plt.xlabel("x")
plt.ylabel("Probability Density")
plt.legend()
plt.grid(True)
plt.show()

Tout était clair ?

Merci pour vos commentaires !

Section 2. Chapitre 1