• ↑
  • Special Distributions

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

  • $f(x)=\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$ for $-\mathrm{\infty }<x<\mathrm{\infty }$

• Properties:

  • Symmetric about the mean
    • Meaning the mean is the median as well
    • So $\mu =0.5$ quantile
    • Mean = median = mode = $\mu$
  • Distribution is completely described by the mean $\mu$ (centre) and variance ${\sigma }^2$ (dispersion)

• Notation

  • If a random variable $X$ follows a normal distribution with mean $\mu$ and variance ${\sigma }^2$, we denote it as: $X\sim N(\mu ,{\sigma }^2)$
  • $N(\mu =-5,{\sigma }^2={0.75}^2)$
  • $N(\mu =1,{\sigma }^2={0.25}^2)$
  • $N(\mu =6,{\sigma }^2=4^2)$
• ```tikz \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/(0.75sqrt(2pi))exp(-((x-(-5))^2)/(20.75^2))};
  \addlegendentry{$N(\mu =-5,{\sigma }^2={0.75}^2)$}

  \addplot[color=red, thick] {1/(0.25sqrt(2pi))exp(-((x-1)^2)/(20.25^2))};
  \addlegendentry{$N(\mu =1,{\sigma }^2={0.25}^2)$}

  \addplot[color=green, thick] {1/(4sqrt(2pi))exp(-((x-6)^2)/(24^2))};
  \addlegendentry{$N(\mu =6,{\sigma }^2=4^2)$}

  \end{axis}
  \end

  \end

STA260

• The distribution of $\overline{X}$ is normal
• $X_1,\dots ,X_n$ are iid from a normal distribution
• $\mathrm{E}[X_i]={\mu }_i$ and $\mathrm{Var}(X_i)={\sigma }_i^2$
• $Y=\sum _{i=1}^n{a}_i{X}_i\sim N(\sum _{i=1}^n{a}_i{\mu }_i,\sum _{i=1}^n{a}_i^2{\sigma }_i^2)$
• If they're iid from a Random Sample from $N(\mu ,{\sigma }^2)$ then $\overline{X}=\frac{1}{n}\sum _{i=1}^n{X}_i\sim N(\mu ,\frac{{\sigma }^2}{n})$