균일분포 (Uniform Distribution)

$$ X \sim U(a , b)$$

• $$ a \in ( \ - \infty \ , b ) $$
• $$ b \in ( a , \ \infty \ ) $$

$$ x \in [ \ a \ , \ b \ ] $$

$$ f(x) = \frac{1}{b-a} $$

$$ F(x) = \frac{x - a}{b - a} $$

$$ E(X) = \frac{a+b}{2} $$

$$ Var(X) = \frac{(b-a)^{2}}{12} $$

$$ \gamma_{1} = 0 $$

$$ \gamma_{2} = - \frac{6}{5} $$

$$ \phi \ (t) = \frac{2}{(b - a) t} \sin \left[ \frac{1}{2} (b-a) t \right] e^{i(a+b)t/2} $$

만약, $a=0 , b=1$일 경우 특성함수는 아래와 같다.

$$ \phi \ (t) = \frac{i - i \cos t + \sin t}{t} $$

$$ M(t) = \frac{e^{tb}-e^{ta}}{t(b-a)} $$

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

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

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

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

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

$$ \mu_{1} = 0 $$

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

$$ \mu_{3} = 0 $$

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

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

$X \sim U(0,\theta)$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