균등분포
\begin{align*}
X &\sim uniform(a, b)
\\[10pt]
f_{X}(x) &= \dfrac {1}{b-a}~~ I(a<x<b)
\\[10pt]
F_{X}(x) &= \dfrac {1}{b-a}\cdot x
\\[10pt]
M_{X}(t) &= \dfrac{e^b-e^a}{b-a}
\end{align*}
확률함수 조건 확인
P~(-\infty \le X \le \infty) = \int_{-\infty}^{\infty}f_X(x)~dx =
\int_{-\infty}^{\infty}
\dfrac{1}{b-a}dx
~=~\dfrac{x}{b-a}
\bigg\rvert_{a}^{b}~=~1
1차 적률 = 평균
\begin{align*}
E(X) &= \int_{-\infty}^{\infty}
x \cdot f_X(x)dx
= \int_{-\infty}^{\infty}
~\dfrac{x}{b-a}dx
\\[20pt]
&= \dfrac{x^2}{2(b-a)}
\bigg\rvert_{a}^{b} =
\dfrac{a+b}{2}
\end{align*}
2차 적률과 분산
\begin{align*}
E(X^2) &=
\int_{-\infty}^{\infty}
x^2 \cdot f_X(x)dx
= \int_{-\infty}^{\infty}
~\dfrac{x^2}{b-a}dx
\\[20pt]
&=\dfrac{x^3}{3(b-a)}
\bigg\rvert_{a}^{b}~~=~
\dfrac{b^2 + ab + a^2}{3}
\end{align*}
\begin{align*}
Var(X) &= E(X^2) - E(X)^2
\\[20pt]
&= \dfrac{b^2 + ab + a^2}{3}
- \bigg\{\dfrac{a+b}{2}\bigg\}^2
\\[20pt]
&=-\dfrac{(b-a)^2}{12}
\end{align*}