====== 균일분포 (Uniform Distribution) ======
===== 정의 =====
===== 표기 =====
 $$ X \sim U(a , b)$$

  * $$ a \in ( \ - \infty \ , b ) $$
  * $$ b \in ( a , \ \infty \ ) $$
===== 받침 =====
 $$ x \in [ \ a \ , \ b \ ] $$
===== 확률밀도함수 =====
 $$ f(x) = \frac{1}{b-a} $$

<plot>
 set title "Uniform Distribution PDF"
 set size 1.0
 set xrange [1:3]
 set yrange [0:1]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "f(x)"

 plot 0.5 title "U(1,3)"
</plot>
===== 누적분포함수 =====
 $$ F(x) = \frac{x - a}{b - a} $$

<plot>
 set title "Uniform Distribution CDF"
 set size 1.0
 set xrange [1:3]
 set yrange [0:1.1]
 set format x "%.1f"
 set format y "%.2f"
 set xlabel "x"
 set ylabel "F(x)"
 set key left

 f(x,a,b) = (x-a)/(b-a)

 plot f(x,1,3) title "U(1,3)"
</plot>
===== 기대값 =====
 $$ E(X) = \frac{a+b}{2} $$
===== 분산 =====
 $$ Var(X) = \frac{(b-a)^{2}}{12} $$
===== 왜도 =====
 $$ \gamma_{1} = 0 $$
===== 첨도 =====
 $$ \gamma_{2} = - \frac{6}{5} $$
===== 특성함수 =====
 $$ \phi \ (t) = \frac{2}{(b - a) t} \sin \left[ \frac{1}{2} (b-a) t \right] e^{i(a+b)t/2} $$

 만약, $a=0 , b=1$일 경우 [[특성함수]]는 아래와 같다.

 $$ \phi \ (t) = \frac{i - i \cos t + \sin t}{t} $$
===== 적률생성함수 =====
 $$ M(t) = \frac{e^{tb}-e^{ta}}{t(b-a)} $$
===== 원적률 =====
 $$ \mu'_{1} = \frac{1}{2} (a+b) $$

 $$ \mu'_{2} = \frac{1}{3} (a^{2} + ab + b^{2}) $$

 $$ \mu'_{3} = \frac{1}{4} (a+b)(a^{2} + b^{2}) $$

 $$ \mu'_{4} = \frac{1}{5} (a^{4} + a^{3} b + a^{2} b^{2} + a b^{3} + b^{4}) $$

 $$ \mu'_{k} = \frac{b^{k+1} - a^{k+1}}{(k+1)(b-a)} $$
===== 중심적률 =====
 $$ \mu_{1} = 0 $$

 $$ \mu_{2} = \frac{1}{12} (b-a)^{2} $$

 $$ \mu_{3} = 0 $$

 $$ \mu_{4} = \frac{1}{80} (b-a)^{4} $$

 $$ \mu_{k} = \frac{(a-b)^{k} + (b-a)^{k}}{2^{k+1} (k+1)} $$
===== 유용한 공식 =====
 $X \sim U(0,\theta)$일 때,

  * $$ E(X_{(k)}) = \frac{k}{n+1} \theta $$
  * $$ E(X_{(k)}^{2}) = \frac{k}{(n+2)^{2}} \theta^{2} $$
  * $$ Var(X_{(k)}) = \frac{k(n-k+1)}{(n+1)^{2} (n+2)} \theta^{2} $$

----
  * [[분포]]
  * [[연속형 분포]]
  * [[순서통계량]]