Introduction

The normal distribution, also known as the Gaussian distribution, plays a pivotal role in probability and statistics. It’s a continuous probability distribution that is symmetric around the mean and follows the famous bell-shaped curve. Understanding the properties and characteristics of the normal distribution is essential in various fields, from finance to natural sciences. In this chapter, we will delve into the key aspects of the normal distribution, its parameters, applications, and how it arises in real-world scenarios.

Properties of Normal Distribution

The normal distribution is defined by two parameters: the mean $(\mu )$ and the standard deviation $(\sigma )$. The mean represents the central location of the distribution, while the standard deviation controls the spread or dispersion of the data points. The probability density function (PDF) of the normal distribution is given by the formula:

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

Here, $x$ represents a random variable, $\mu$ is the mean, $\sigma$ is the standard deviation, $e$ is the base of the natural logarithm, and $\pi$ is the constant.

Standard Normal Distribution

The standard normal distribution, also known as the Z-distribution, is a special case of the normal distribution with a mean $(\mu )$ of 0 and a standard deviation $(\sigma )$ of 1 . It’s often used for standardizing data to facilitate comparisons and calculations. The formula for the standard normal distribution is:

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

Applications and Examples

The normal distribution arises naturally in many real-world scenarios. One classic example is the distribution of heights in a population. In finance, stock returns often follow a normal distribution, which forms the basis for various risk and portfolio management models. Another application is quality control, where measurements of products’ dimensions are often normally distributed.

Empirical Rule and Percentiles

The normal distribution follows the empirical rule (also known as the 68-95-99.7 rule), which states that approximately 68% of data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.