Normal Distribution

The Normal (Gaussian) Distribution

The normal distribution is the most important continuous distribution in probability and statistics. Its PDF is:

$$f_X(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

Parameters

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

Notation

We write $X \sim N(\mu, \sigma^2)$ to indicate $X$ follows a normal distribution with mean $\mu$ and variance $\sigma^2$.

Properties

Shape

• Symmetric around the mean $\mu$
• Bell-shaped curve
• The peak is at $x = \mu$
• The curve approaches 0 as $x \to \pm\infty$

Spread

• About 68% of values fall within $\mu \pm \sigma$
• About 95% of values fall within $\mu \pm 2\sigma$
• About 99.7% of values fall within $\mu \pm 3\sigma$

This is the empirical rule or 68-95-99.7 rule.

Standard Normal Distribution

The standard normal has $\mu = 0$ and $\sigma = 1$:

$$\phi(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}$$

We use $Z$ to denote a standard normal random variable.

Standardization

Any normal $X \sim N(\mu, \sigma^2)$ can be converted to standard normal:

$$Z = \frac{X - \mu}{\sigma}$$

Then $Z \sim N(0, 1)$.

Cumulative Distribution Function

The CDF of the standard normal is denoted $\Phi(z)$:

$$\Phi(z) = P(Z \leq z) = \int_{-\infty}^z \phi(t) \, dt$$

Properties

• $\Phi(0) = 0.5$ (by symmetry)
• $\Phi(-z) = 1 - \Phi(z)$
• Values are tabulated or computed numerically (no closed form)

Why Is Normal Distribution Important?

Central Limit Theorem

The sum (or average) of many independent random variables tends to be normally distributed, regardless of the original distribution.

Natural Phenomena

Many natural measurements are approximately normal:

• Heights, weights
• Measurement errors
• Test scores
• Physical quantities

Statistical Inference

Many statistical methods assume normality:

• Confidence intervals
• Hypothesis tests
• Regression analysis

Calculating Probabilities

Example: Finding $P(a \leq X \leq b)$

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

1. Standardize: $Z = \frac{X - \mu}{\sigma}$
2. Convert bounds: $z_a = \frac{a - \mu}{\sigma}$, $z_b = \frac{b - \mu}{\sigma}$
3. Use standard normal table or calculator:

   $$P(a \leq X \leq b) = \Phi(z_b) - \Phi(z_a)$$

Example

For $X \sim N(100, 15^2)$, find $P(85 \leq X \leq 115)$:

1. $z_1 = \frac{85 - 100}{15} = -1$
2. $z_2 = \frac{115 - 100}{15} = 1$
3. $P(-1 \leq Z \leq 1) = \Phi(1) - \Phi(-1) = 0.8413 - 0.1587 = 0.6826$

About 68% of values fall within one standard deviation of the mean.