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.

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.

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).

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.

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.

6. Examples

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

7. Applications

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

8. Visualization (Conceptual)

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

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.