Normal Distribution

A type of Continuous Probability Distributions $\mathcal{N}(\mu ,{\sigma }^2)$ Best used to model an average of independent RVs. This distribution can be modelled with the Density Function

PDF

$f(x)=\frac{1}{\sqrt{2\pi \sigma }}\exp \left\{-\frac{1}{2}(\frac{x-\mu }{\sigma }{)}^2\right\},x\in \mathbb{R}$ With:

• $\sigma$ as Standard Deviation
• $\mu$ as Mean
• $\text{exp}$ is Exponential Function

CDF

$$F(x)=\frac{1}{2}[1+\text{erf}(\frac{x-\mu }{\sigma \sqrt{2}})]$$

Expectation

$E(X)=\mu$

Variance

$V(X)={\sigma }^2$

MGF

$$\exp \{ut+\frac{1}{2}{\sigma }^2{t}^2\}$$

Concepts

• Standard Normal Distribution
• Normal Distribution Standard Deviation Approximation
• 68,95,99.7 Rule