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