Normal Distribution

probability-theory

Definition

Normal Distribution

The normal distribution, often called the Gaussian distribution, is a continuous probability distribution characterised by its bell-shaped curve that is symmetric about the mean. It is a fundamental distribution in statistics, describing how values of a variable are distributed, with most values clustering around a central mean and fewer occurring as you move away, conforming to a specific mathematical formula.

Density Function

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

Meaning of $\sigma$ for Density Function

Meaning of $\sigma$

The smaller sigma is, the smaller the width of the bell is.

Meaning of $\mu$ for Density Function

Meaning of $\mu$

$\mu$ shifts the center of the bell along the x axis.

Distribution Function

$$F(x)={\int }_{-\infty }^{x}f(x)dx$$

Meaning of $\sigma$ for Distribution Function

$$\begin{array}{rl}\text{red}:\sigma & =2\\ \text{green}:\sigma & =5\\ \text{black}:\sigma & =8\end{array}$$

$\sigma$-Environments

$$\begin{array}{rl}P(\mu -\sigma \le X\le \mu +\sigma )& \approx 0.683\\ P(\mu -2\sigma \le X\le \mu +2\sigma )& \approx 0.954\\ P(\mu -3\sigma \le X\le \mu +3\sigma )& \approx 0.997\end{array}$$

Standard Normal Distribution

A normal distribution is called standard normal distribution if $\sigma =1$ and $\mu$ = 0. Hence, the density function looks as follows:

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

Exam Questions

Exam Question: $\sigma$ and $\mu$ missing

Instructions:

Assume that the length of planks are approximately normal distributed.

• 24% of the planks are longer than 4.5 meters
• 88% of the planks are longer than 3.2 meters Find $\mu$ and $\sigma$.

Solution:

1. Find $z_1$ such that: ${\int }_{z_1}^{\infty }{f}_n(x)dx=24\%$
2. Find $z_2$ such that ${\int }_{z_2}^{\infty }{f}_n(x)dx=88\%$
3. Find $\mu$ and $\sigma$ by solving the system of equations:

$$\{\begin{array}{ll}{z}_1& =\frac{4.5-\mu }{\sigma }\\ z_2& =\frac{3.2-\mu }{\sigma }\end{array}\Rightarrow \mu \approx 4.012\wedge \sigma \approx 0.691$$

Validation:

$$\begin{array}{rl}{\int }_{4.5}^{\infty }f(x)dx& =24\%\\ & \\ {\int }_{3.2}^{\infty }f(x)dx& =88\%\end{array}$$