n回の事象が発生するまでの間隔の確率密度分布 (ガンマ分布)

\(n\)回の事象が発生するまでの間隔(時間)の確率\(Y_n\)について考える． $$ \begin{eqnarray} F_n(t) &=&P(Y_n\leq t) \;\cdots\;F_nをY_nの累積分布凾数(cumulative\;distribution\;function, CDF)とする．(少なくともtまでにn回の事象が発生している)\\ &=&P(X_t\geq n) \;\cdots\;n回以上の事象が発生するのが時刻t以降になる確率とも言える\\ &=&\sum_{x=n}^\infty P_o(x;\lambda t) \;\cdots\;事象の発生回数の確率を\lambda tをパラメータとするポアソン分布とする．ポアソン分布: P_o(x;\lambda)= \mathrm{e}^{-\lambda} \frac{\left(\lambda\right)^x}{x!}\\ &=&\sum_{x=n}^\infty\mathrm{e}^{-\lambda t} \frac{\left(\lambda t\right)^x}{x!}\;\cdots\;ポアソン分布は期間内で何回起こるか？の確率分布(指数分布は次の発生までの期間の確率分布)\\ &=&1-\sum_{x=0}^{n-1}\mathrm{e}^{-\lambda t} \frac{\left(\lambda t\right)^x}{x!} \;\cdots\;時刻tまでにn回未満の事象しか発生しない場合の余事象\\ &=&1-\mathrm{e}^{-\lambda t}\left\{ \frac{\left(\lambda t\right)^0}{0!} + \frac{\left(\lambda t\right)^1}{1!} + \frac{\left(\lambda t\right)^2}{2!} + \dots + \frac{\left(\lambda t\right)^\left(n-2\right)}{\left(n-2\right)!} + \frac{\left(\lambda t\right)^\left(n-1\right)}{\left(n-1\right)!} \right\}\\ &=&1-\mathrm{e}^{-\lambda t}A \;\cdots\;A=\left\{ \frac{\left(\lambda t\right)^0}{0!} + \frac{\left(\lambda t\right)^1}{1!} + \frac{\left(\lambda t\right)^2}{2!} + \dots + \frac{\left(\lambda t\right)^\left(n-2\right)}{\left(n-2\right)!} + \frac{\left(\lambda t\right)^\left(n-1\right)}{\left(n-1\right)!} \right\}\\ Y_n=\frac{\partial F_n(t)}{\partial t} &=&0-\left[(\mathrm{e}^{-\lambda t})'A +\mathrm{e}^{-\lambda t}(A)' \right] \;\cdots\;\href{https://shikitenkai.blogspot.com/2020/02/blog-post.html}{(fg)'=f'g+fg'} \;\cdots\;累積分布凾数の微分は確率密度凾数(probability\;density\;function、PDF)\\ (\mathrm{e}^{-\lambda t})'A &=&\mathrm{e}^{-\lambda t}(-\lambda)A\\ &=&\mathrm{e}^{-\lambda t}(-\lambda)\left\{ \frac{\left(\lambda t\right)^0}{0!} + \frac{\left(\lambda t\right)^1}{1!} + \frac{\left(\lambda t\right)^2}{2!} + \dots + \frac{\left(\lambda t\right)^\left(n-2\right)}{\left(n-2\right)!} + \frac{\left(\lambda t\right)^\left(n-1\right)}{\left(n-1\right)!} \right\}\\ &=&\mathrm{e}^{-\lambda t}\left\{ - \frac{\lambda}{0!} - \frac{\lambda^2 t}{1!} - \frac{\lambda^3 t^2}{2!} - \dots - \frac{\lambda^\left(n-1\right) t^\left(n-2\right)}{\left(n-2\right)!} - \frac{\lambda^n t^\left(n-1\right)}{\left(n-1\right)!} \right\}\\ \\ \mathrm{e}^{-\lambda t}(A)' &=&\mathrm{e}^{-\lambda t}\left(\left\{ \frac{\left(\lambda t\right)^0}{0!} + \frac{\left(\lambda t\right)^1}{1!} + \frac{\left(\lambda t\right)^2}{2!} + \dots + \frac{\left(\lambda t\right)^\left(n-2\right)}{\left(n-2\right)!} + \frac{\left(\lambda t\right)^\left(n-1\right)}{\left(n-1\right)!} \right\}\right)'\\ &=&\mathrm{e}^{-\lambda t}\left\{ \left(\frac{1}{0!}\right)' + \left(\frac{\lambda t}{1!}\right)' + \left(\frac{\lambda^2 t^2}{2!}\right)' + \dots + \left(\frac{\lambda^\left(n-2\right) t^\left(n-2\right)}{\left(n-2\right)!}\right)' + \left(\frac{\lambda^\left(n-1\right) t^\left(n-1\right)}{\left(n-1\right)!}\right)' \right\}\\ &=&\mathrm{e}^{-\lambda t}\left\{ 0 + \frac{\lambda}{1!} + \frac{2\lambda^2 t}{2!} + \dots + \frac{(n-2)\lambda^{(n-2)} t^\left(n-3\right)}{\left(n-2\right)!} + \frac{(n-1)\lambda^{(n-1)} t^\left(n-2\right)}{\left(n-1\right)!} \right\}\\ &=&\mathrm{e}^{-\lambda t}\left\{ \frac{\lambda}{1!} + \frac{\lambda^2 t}{1!} + \dots + \frac{\lambda^{(n-2)} t^\left(n-3\right)}{\left(n-3\right)!} + \frac{\lambda^{(n-1)} t^\left(n-2\right)}{\left(n-2\right)!} \right\} \;\cdots\;\frac{n}{n!}=\frac{n}{n\times(n-1)\times\dots\times 1}=\frac{1}{(n-1)\times\dots\times 1}=\frac{1}{(n-1)!}\\ (\mathrm{e}^{-\lambda t})'A+\mathrm{e}^{-\lambda t}(A)' &=&\mathrm{e}^{-\lambda t}\left\{ - \frac{\lambda^1}{0!} - \frac{\lambda^2 t}{1!} - \frac{\lambda^3 t^2}{2!} - \dots - \frac{\lambda^\left(n-1\right) t^\left(n-2\right)}{\left(n-2\right)!} - \frac{\lambda^n t^\left(n-1\right)}{\left(n-1\right)!} \right\} + \mathrm{e}^{-\lambda t}\left\{ \frac{\lambda}{1!} + \frac{\lambda^2 t}{1!} + \dots + \frac{\lambda^{(n-2)} t^\left(n-3\right)}{\left(n-3\right)!} + \frac{\lambda^{(n-1)} t^\left(n-2\right)}{\left(n-2\right)!} \right\}\\ &=&\mathrm{e}^{-\lambda t}\left\{ \left(- \frac{\lambda}{0!}+ \frac{\lambda}{1!}\right) + \left(- \frac{\lambda^2 t}{1!}+ \frac{\lambda^2 t}{1!}\right) + \dots + \left(- \frac{\lambda^\left(n-1\right) t^\left(n-2\right)}{\left(n-2\right)!}+ \frac{\lambda^{(n-1)} t^\left(n-2\right)}{\left(n-2\right)!}\right) - \frac{\lambda^n t^\left(n-1\right)}{\left(n-1\right)!} \right\}\\ &=&\mathrm{e}^{-\lambda t}\left\{ - \frac{\lambda^n t^\left(n-1\right)}{\left(n-1\right)!} \right\}\\ Y_n=\frac{\partial F_n(t)}{\partial t} &=&-\left[(\mathrm{e}^{-\lambda t})'A +\mathrm{e}^{-\lambda t}(A)' \right]\\ &=&-\left[\mathrm{e}^{-\lambda t}\left\{ - \frac{\lambda^n t^\left(n-1\right)}{\left(n-1\right)!} \right\}\right]\\ &=&\mathrm{e}^{-\lambda t}\frac{\lambda^n t^\left(n-1\right)}{\left(n-1\right)!}\\ &=&\mathrm{e}^{-\beta t}\frac{\beta^\alpha t^\left(\alpha-1\right)}{\left(\alpha-1\right)!} \;\cdots\;\alpha=n, \beta=\lambdaとする.\\ &=&\frac{\mathrm{e}^{-\beta t}}{\left(\alpha-1\right)!}\beta^\alpha t^\left(\alpha-1\right)\\ &=&\frac{\mathrm{e}^{-\beta t}}{\Gamma(\alpha)}\beta^\alpha t^\left(\alpha-1\right) \;\cdots\;ガンマ分布, \Gamma(n)=(n-1)!\\ \end{eqnarray} $$ \(\alpha=1\)の時，指数分布となる(事象と事象の間隔の確率分布)． $$ \begin{eqnarray} \frac{\mathrm{e}^{-\beta t}}{\Gamma(\alpha)}\beta^\alpha t^\left(\alpha-1\right) &=&\frac{\mathrm{e}^{-\beta t}}{\Gamma(1)}\beta^1 t^\left(1-1\right)\;\cdots\;\alpha=1\\ &=&\frac{\mathrm{e}^{-\beta t}}{1}\beta t^\left(0\right)\;\cdots\;\Gamma(n)=(n-1)!,\Gamma(1)=(1-1)!=0!=1\\ &=&\mathrm{e}^{-\beta t}\beta\;\cdots\;t^0=1\\ &=&\href{https://shikitenkai.blogspot.com/2020/05/blog-post_0.html}{\lambda\mathrm{e}^{-\lambda t}}\;\cdots\;\beta=\lambda\\ \end{eqnarray} $$