12. Normal Distribution

$\gdef \N{\mathbb{N}} \gdef \Z{\mathbb{Z}} \gdef \Q{\mathbb{Q}} \gdef \R{\mathbb{R}} \gdef \C{\mathbb{C}} \gdef \setcomp#1{\overline{#1}} \gdef \sseq{\subseteq} \gdef \pset#1{\mathcal{P}(#1)} \gdef \covariant{\operatorname{Cov}} \gdef \of{\circ} \gdef \p{^{\prime}} \gdef \pp{^{\prime\prime}} \gdef \ppp{^{\prime\prime\prime}} \gdef \pn#1{^{\prime\times{#1}}} $

Standard Normal Distribution¶

Definition¶

\(Z=N(0,1)\), where \(0=\mu=E[Z]\), and \(1=V(Z)=\sigma^2\).

\(f(x)=\frac1{\sqrt{2\pi}}e^{\frac{-x^2}2}\)

\(F(x)=P(Z\le X)=\int_{-\infty}^xf(t)dt=\Phi(x)\)
Use chart for \(\Phi\), because we don’t have a formula.

General Normal Distribution¶

Consider the function \(f(x)=\frac1{\sqrt{2\pi}}\)

if \(X=N(\mu,\sigma^2)\), \(Z=\frac{X-a}{b}\sim N(0,1)\)