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11. Continuous Distributions

$\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}}} $

Uniform Distribution

Definition

\(X\sim U(a,b)\) means that \(X\) can take any value in the interval \((a,b)\), with probability equally dense everywhere in the interval density function.

For example, take the following density function: \(f(x)=\begin{cases}0&b>x<a\\\frac1{b-a}&a<x<b\end{cases}\)

Excalidraw Excalidraw

The cumulative distribution function is: \(F(x)=\begin{cases}0&x<a\\\int_a^x\frac1{b-a}dt=\frac{t}{b-a}\bigr|_a^x=\frac{x-a}{b-a}&a<x<b\\1&x>b\end{cases}\)

\[\begin{aligned}F(x)&=\begin{cases}0&x<a\\\frac{x-a}{b-a}&a<x<b\\1&x>b\end{cases}\\E[X]&=\frac{a+b}2\\V(X)&=\frac{(b-a)^2}{12}\\\sigma(X)&=\frac{b-a}{2\sqrt{3}}\end{aligned}\]

Exponential Distribution

\(X\sim Exp(\lambda)\)

\[\begin{aligned}f(x)&=\cases{\lambda e^{-\lambda x} &{$x>0$}\\0&$x\le0$}\\F(x)&=\cases{1-e^{-\lambda x} &{$x>0$}\\0&$x\le0$}\\E[X]&=\frac1\lambda\\V(X)&=\frac1{\lambda^2}\end{aligned}\]