삼각분포 (Triangular Distribution)

단, $a \leq x \leq c \ , \ a \leq m \leq c$

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

$$ p(x) = \left\{ \begin{displaymath}\begin{split} \frac{2(x - a)}{(b - a)(m - a)} & \ \ \ \ (a \leq x \leq c) \\ \frac{2(b - x)}{(b - a)(b - m)} & \ \ \ \ (c < x \leq b) \end{split}\end{displaymath} $$ ===== 누적분포함수 ===== $$ F(x) = \left\{ \begin{displaymath}\begin{split} \frac{2(x - a)^{2}}{(b - a)(m - a)} & \ \ \ \ (a \leq x \leq c) \\ 1 - \frac{2(b - x)^{2}}{(b - a)(b - m)} & \ \ \ \ (c < x \leq b) \end{split}\end{displaymath} $$ ===== 기대값 ===== $$ E(X) = \frac{a + m + b}{3} $$ ===== 최빈값 ===== $$ Mo = m $$ ===== 분산 ===== $$ Var(X) = \frac{1}{18} (a^{2} + m^{2} + b^{2} - am - ab - mb) $$ ===== 왜도 ===== $$ \gamma_{1} = \frac{\sqrt{2} \ (a+b-2m)(2a-b-m)(a-2b+m)}{5(a^{2} + b^{2} + c^{2} - ab - am - mb)^{3/2}} $$ ===== 첨도 ===== $$ \gamma_{2} = - \frac{3}{5} $$ ===== 원적률 ===== $$ \mu'_{2} = \frac{1}{6} (a^{2} + m^{2} + b^{2} + ab + am + mb) $$ $$ \mu'_{3} = \frac{1}{10} (a^{3} + m^{3} + b^{3} + a^{2}b + a^{2}m + b^{2}a + b^{2}m + m^{2}a + m^{2}b + amb) $$ ===== 중심적률 ===== $$ \mu_{2} = \frac{1}{18} (a^{2} + m^{2} + b^{2} - ab - am - mb) $$ $$ \mu_{3} = - \frac{1}{270} (a + b - 2m) (a + m -2b) (b + m -2a) $$ $$ \mu_{4} = \frac{1}{135} (a^{2} + m^{2} + c^{2} - ab - am - mb)^{2} $$

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