Normal Distribution

STA256

Bell curve distribution
Very useful because this models a lot of natural real life processes.
Random Variable $X$ has a normal distribution $⟺$
Probability Density Function
- $f (x) = \frac{1}{\sqrt{2 π σ^{2}}} e^{- \frac{(x - μ)^{2}}{2 σ^{2}}}$ for $- \infty < x < \infty$
Properties:
- Symmetric about the mean
  - Meaning the mean is the median as well
  - So $μ = 0.5$ quantile
  - Mean = median = mode = $μ$
- Distribution is completely described by the mean $μ$ (centre) and variance $σ^{2}$ (dispersion)
Notation
- If a random variable $X$ follows a normal distribution with mean $μ$ and variance $σ^{2}$ , we denote it as: $X \sim N (μ, σ^{2})$
- $N (μ = - 5, σ^{2} = {0.75}^{2})$
- $N (μ = 1, σ^{2} = {0.25}^{2})$
- $N (μ = 6, σ^{2} = 4^{2})$
```
\usepackage{pgfplots} 
\pgfplotsset{compat=1.16}
\begin{document} 

\begin{tikzpicture} 
\begin{axis}[ 
    xlabel={$\mu$}, 
    ylabel={$P(x)$}, 
    domain=-15:20, 
    samples=100, 
    legend pos=outer north east, 
    title={Normal Distributions}, 
] 

\addplot[color=blue, thick] {1/(0.75*sqrt(2*pi))*exp(-((x-(-5))^2)/(2*0.75^2))}; 
\addlegendentry{$N(\mu=-5, \sigma^2=0.75^2)$} 

\addplot[color=red, thick] {1/(0.25*sqrt(2*pi))*exp(-((x-1)^2)/(2*0.25^2))}; 
\addlegendentry{$N(\mu=1, \sigma^2=0.25^2)$} 

\addplot[color=green, thick] {1/(4*sqrt(2*pi))*exp(-((x-6)^2)/(2*4^2))}; 
\addlegendentry{$N(\mu=6, \sigma^2=4^2)$} 

\end{axis} 
\end{tikzpicture} 

\end{document} 
```

The distribution of $\bar{X}$ is normal
$X_{1}, \dots, X_{n}$ are iid from a normal distribution
$E [X_{i}] = μ_{i}$ and $Var (X_{i}) = σ_{i}^{2}$
$Y = \sum_{i = 1}^{n} a_{i} X_{i} \sim N (\sum_{i = 1}^{n} a_{i} μ_{i}, \sum_{i = 1}^{n} a_{i}^{2} σ_{i}^{2})$
If they're iid from a Random Sample from $N (μ, σ^{2})$ then $\bar{X} = \frac{1}{n} \sum_{i = 1}^{n} X_{i} \sim N (μ, \frac{σ^{2}}{n})$