Normal Distribution

Overview

The normal distribution (Gaussian distribution) is the most important continuous probability distribution. It describes many natural phenomena and is fundamental to statistical inference.

Probability Density Function

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

Parameters

• $\mu$ (mu): Mean (center of the distribution)
• $\sigma$ (sigma): Standard deviation (spread)

Notation

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

or

$$X \sim \text{Normal}(\mu, \sigma)$$

Properties

Property	Value
Mean	$\mu$
Median	$\mu$
Mode	$\mu$
Variance	$\sigma^2$
Standard Deviation	$\sigma$
Skewness	0
Kurtosis (excess)	0

The Bell Curve

        ╭───────╮
       ╱         ╲
      ╱           ╲
     ╱             ╲
────╱───────────────╲────
         μ
    ←  σ  →  ←  σ  →

• Symmetric about the mean
• Asymptotic to x-axis (never touches)
• Total area under curve = 1

The Empirical Rule (68-95-99.7)

For normally distributed data:

$$P(\mu - \sigma < X < \mu + \sigma) \approx 68\%$$ $$P(\mu - 2\sigma < X < \mu + 2\sigma) \approx 95\%$$ $$P(\mu - 3\sigma < X < \mu + 3\sigma) \approx 99.7\%$$

Range	Approximate %
$\mu \pm 1\sigma$	68.27%
$\mu \pm 2\sigma$	95.45%
$\mu \pm 3\sigma$	99.73%

Standard Normal Distribution

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

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

Standardization

Convert any normal to standard normal:

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

Convert back:

$$X = \mu + Z \cdot \sigma$$

Working with Normal Distributions

Finding Probabilities

1. Standardize: $Z = (X - \mu) / \sigma$
2. Use Z-table or calculator
3. Adjust for direction (left, right, between)

Finding Values from Probabilities

1. Find $Z$ from the probability
2. Convert: $X = \mu + Z \cdot \sigma$

Z-Table Reference

$Z$	$P(Z < z)$	$Z$	$P(Z < z)$
-3.0	0.0013	0.0	0.5000
-2.0	0.0228	1.0	0.8413
-1.0	0.1587	2.0	0.9772
-0.5	0.3085	3.0	0.9987

Examples

Example 1: Finding Probability

Test scores are $N(70, 10)$. Find $P(X < 85)$.

$$Z = \frac{85 - 70}{10} = 1.5$$ $$P(Z < 1.5) = 0.9332$$

Example 2: Greater Than

Same distribution. Find $P(X > 60)$.

$$Z = \frac{60 - 70}{10} = -1.0$$ $$P(Z > -1.0) = 1 - P(Z < -1.0) = 1 - 0.1587 = 0.8413$$

Example 3: Between Two Values

Find $P(65 < X < 80)$.

$$Z_1 = \frac{65 - 70}{10} = -0.5$$ $$Z_2 = \frac{80 - 70}{10} = 1.0$$ $$P(-0.5 < Z < 1.0) = P(Z < 1.0) - P(Z < -0.5) = 0.8413 - 0.3085 = 0.5328$$

Example 4: Finding a Percentile

Find the score at the 90th percentile.

$$P(Z < z) = 0.90 \quad \Rightarrow \quad z \approx 1.28$$ $$X = \mu + Z \cdot \sigma = 70 + 1.28 \times 10 = 82.8$$

Linear Combinations

If $X \sim N(\mu_X, \sigma_X^2)$ and $Y \sim N(\mu_Y, \sigma_Y^2)$ are independent:

$$aX + bY \sim N(a\mu_X + b\mu_Y, a^2\sigma_X^2 + b^2\sigma_Y^2)$$

Why So Important?

1. Many natural phenomena are approximately normal
2. Central Limit Theorem: sample means are normally distributed
3. Basis for many statistical tests
4. Convenient mathematical properties