'''Uniform Random Variable'''
AKA '''균일확률변수'''

= Discrete =
'''이산균등확률변수'''

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

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

= Continuous =
'''연속균등확률변수'''

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

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

= 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'''을 $U\in[a,b]$ 로 두면,

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

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

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

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


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Source: Leon-Garcia Table 3.1 (discrete), Table 4.1 (continuous)
Up:
 [[이산확률변수,discrete_RV]]
 [[연속확률변수,continuous_RV]]