균등확률변수,uniform_RV

Uniform Random Variable
AKA 균일확률변수

Discrete

이산균등확률변수

$\displaystyle S_X=\{1,2,\cdots,L\}$
$\displaystyle p_k=\frac1L$
$\displaystyle k=1,2,\cdots,L$

$\displaystyle E[X]=\frac{L+1}{2}$
$\displaystyle V[X]=\frac{L^2-1}{12}$

Continuous

연속균등확률변수

$\displaystyle S_X=[a,b]$
$\displaystyle f_X(x)=\frac1{b-a}$
$\displaystyle a\le x\le b$

$\displaystyle E[X]=\frac{a+b}2$
$\displaystyle V[X]=\frac{(b-a)^2}{12}$

from 경북대강의

// http://www.kocw.net/home/search/kemView.do?kemId=1279832 12. Function of Random Variable, The Expected Value of Random Variables

Uniform random variable$\displaystyle U\in[a,b]$ 로 두면,

PDF:
$\displaystyle f_U(x)=\begin{cases}\frac1{b-a}&a\le x\le b\\0&\textrm{otherwise}\end{cases}$

CDF:
$\displaystyle F_U(x)=\begin{cases}P(a\le U\le x)=\int_a^x\frac1{b-a}dt=\frac{x-a}{b-a}&(a\le x\le b)\\0&(x<a)\\1&(x>b)\end{cases}$

$\displaystyle E(U)=\int_a^b u\frac1{b-a} du = \frac{a+b}2$

$\displaystyle Var(U)=E(U^2)-E(U)^2$
$\displaystyle =\int_a^b u^2 \frac1{b-a} du - \left(\frac{a+b}2\right)^2 = \frac{(a-b)^2}{12}$



Source: Leon-Garcia Table 3.1 (discrete), Table 4.1 (continuous)
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