지수 ⇒ 감마

\begin{align*} F_Y(y) &= P(Y \le y) \\[10pt] &= 1 - P(Y \le y) = 1 - P(X = 0) \\[10pt] &= 1 - \frac {e^{-\lambda y}~(\lambda y)^0}{0!} = 1 - e^{-\lambda y} \\[15pt] f_Y(y) &= \dfrac{d}{dy}F_Y(y) = 1 - e^{-\lambda y} \end{align*}

\begin{align*} F_W(w) &= P(W \le w) = 1 - P(W > w) \\[10pt] &= 1 - P(X \le a-1) \\[10pt] &= 1 - \sum_{i=0}^{a-1}~ \frac{e^{-\lambda w} ~(\lambda w)^{k}}{k!} \end{align*}

\begin{align*} &f_W(w) = \dfrac{d}{dw}F_W(w) = \dfrac{d}{dw} \bigg\{ 1 - \sum_{i=0}^{a-1}~ \frac{e^{-\lambda w} ~(\lambda w)^{k}}{k!} \bigg\} \\[20pt] &= -\dfrac{d}{dw} \bigg\{- \sum_{i=0}^{a-1}~ \frac{e^{-\lambda w} ~(\lambda w)^{k}}{k!} \bigg\} \\[20pt] &= -\dfrac{d}{dw} \bigg\{e^{-\lambda w}+ \sum_{i=1}^{a-1}~ \frac{e^{-\lambda w} ~(\lambda w)^{k}}{k!} \bigg\} \\[20pt] &= - \bigg\{ -\lambda e^{-\lambda w}+ \sum_{i=1}^{a-1}~ \dfrac{d}{dw} \frac{e^{-\lambda w} ~(\lambda w)^{k}}{k!} \bigg\} \\[20pt] \end{align*}

\begin{align*} &= - \bigg[ -\lambda e^{-\lambda w} + \sum_{i=1}^{a-1}~ \dfrac{1}{k!} \cdot \Big\{ (-\lambda)e^{-\lambda k} (\lambda k)^k + k\lambda e^{-\lambda k} (\lambda w)^{k-1} \Big\} \bigg] \\[20pt] &= - \Bigg[ -\lambda e^{-\lambda w} + \sum_{i=1}^{a-1}~ \bigg\{ \dfrac{e^{-\lambda w}}{k!} (\lambda w)^{k-1} (-\lambda^2w + k\lambda) \bigg\} \Bigg] \\[20pt] &= \lambda e^{-\lambda w} \bigg\{ 1-\dfrac{(\lambda w)^0}{1!}\cdot1 + \dfrac{(\lambda w)^{\alpha-1}}{(\alpha-1)!} \bigg\} \\[20pt] &= \dfrac{1} {(\alpha-1)!~ \big( \frac{1}{\lambda} \big) ^{\alpha} }\cdot w^{\alpha-1} \cdot \exp ^{-w \div \frac{1}{\lambda}} \end{align*}

f_W(w) = \dfrac{1} {(\alpha-1)!~ \beta^{\alpha} } \cdot w^{\alpha-1} \cdot e~ ^{\normalsize\frac{w}{\lambda}}

f_X(x) = \dfrac{1} {\Gamma({\alpha})~ \beta^{\alpha} } \cdot x^{\alpha-1} \cdot e~ ^{\normalsize\frac{x}{\lambda}}