Normal Distribution

1. Definition

• The normal distribution (also called Gaussian distribution) is a continuous probability distribution shaped like a bell curve.
• Many natural and social phenomena (heights, test scores, measurement errors) approximately follow it.

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2. Probability Density Function (PDF)

For a random variable $X \sim N(\mu, \sigma^2)$:

$f(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp\!\Bigg(-\frac{(x-\mu)^2}{2\sigma^2}\Bigg)$

Where:

• $\mu$ = mean (center of the distribution).
• $\sigma^2$ = variance (spread).
• $\sigma$ = standard deviation.

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3. Properties

• Symmetry: Bell-shaped and symmetric around the mean $\mu$.
• Mean, Median, Mode: all equal to $\mu$.
• Spread: Controlled by $\sigma$. Larger $\sigma$ = flatter and wider curve.
• Total probability = 1 (area under curve).

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4. Standard Normal Distribution

• Special case: $\mu = 0, \sigma = 1$.
• Denoted $Z \sim N(0,1)$.
• Used with z-scores: $z = \frac{x – \mu}{\sigma}$​ to standardize data and use standard normal tables.

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5. Probability Ranges (Empirical Rule: 68–95–99.7 Rule)

• About 68% of values lie within ±1σ of the mean.
• About 95% within ±2σ.
• About 99.7% within ±3σ.

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6. Examples

• Human height: ~ Normal(170, 10²).
• Test scores (after standardization).
• Measurement errors in experiments.

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7. Applications

• Statistics: hypothesis testing, confidence intervals, regression assumptions.
• Forecasting: modeling demand, returns, or prediction errors.
• Machine Learning: Gaussian likelihoods, loss functions, Bayesian priors.

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8. Visualization (Conceptual)

• Small σ → tall, narrow curve.
• Large σ → flat, wide curve.
• Symmetric about $\mu$.

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Summary:
The Normal Distribution $N(\mu, \sigma^2)$ is a bell-shaped, symmetric distribution described by mean $\mu$ and variance $\sigma^2$. It is central in probability and statistics due to the Central Limit Theorem, which says sums/averages of many random variables tend to be approximately normal.