離散型確率変数(discrete random variable) の一様分布(uniform distribution)の期待値(expected value)

$$\begin{array}{rcl}{\displaystyle M_X^{(m)}(0)}& \equiv & \frac{{\mathrm{d}}^m}{{\mathrm{d}}^{m}t}{M}_X(t){|}_{t=0}\\ & =& {\displaystyle E[X^m{\mathrm{e}}^{tX}]{|}_{t=0}}\\ & =& {\displaystyle E[X^m]}\end{array}$$

積率母凾数の一階微分

$$\begin{array}{rcl}{\displaystyle M_X^{(1)}}& =& {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\left\{{\displaystyle \frac{1}{n}\frac{{\mathrm{e}}^t({\mathrm{e}}^{nt}-1)}{({\mathrm{e}}^t-1)}{\displaystyle }}\right\}}\\ & =& {\displaystyle \frac{1}{n}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\left\{{\displaystyle \frac{{\mathrm{e}}^t({\mathrm{e}}^{nt}-1)}{({\mathrm{e}}^t-1)}{\displaystyle }}\right\}}}\\ & =& {\displaystyle \frac{1}{n}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\left\{{\displaystyle {\mathrm{e}}^t({\mathrm{e}}^{nt}-1)({\mathrm{e}}^t-1{)}^{-1}{\displaystyle }}\right\}}}\\ & =& {\displaystyle \frac{1}{n}\{({\mathrm{e}}^t{)}^{\prime }({\mathrm{e}}^{nt}-1)({\mathrm{e}}^t-1{)}^{-1}+{\mathrm{e}}^t({\mathrm{e}}^{nt}-1{)}^{\prime }({\mathrm{e}}^t-1{)}^{-1}+{\mathrm{e}}^t({\mathrm{e}}^{nt}-1)(({\mathrm{e}}^t-1{)}^{-1}{)}^{\prime }\}}\\ & & \dots (uvw{)}^{\prime }=u^{\prime }(vw)+u(vw{)}^{\prime }=u^{\prime }(vw)+u(v^{\prime }w+v{w}^{\prime })=u^{\prime }vw+u{v}^{\prime }w+uv{w}^{\prime }\\ & =& {\displaystyle \frac{1}{n}\{{\mathrm{e}}^t({\mathrm{e}}^{nt}-1)({\mathrm{e}}^t-1{)}^{-1}+{\mathrm{e}}^t(n{\mathrm{e}}^{nt})({\mathrm{e}}^t-1{)}^{-1}+{\mathrm{e}}^t({\mathrm{e}}^{nt}-1)(-{\mathrm{e}}^t({\mathrm{e}}^t-1{)}^{-2})\}}\\ & =& {\displaystyle \frac{1}{n}\{\frac{{\mathrm{e}}^t({\mathrm{e}}^{nt}-1)}{({\mathrm{e}}^t-1)}+\frac{{\mathrm{e}}^t(n{\mathrm{e}}^{nt})}{({\mathrm{e}}^t-1)}+\frac{{\mathrm{e}}^t{\mathrm{e}}^t({\mathrm{e}}^{nt}-1)}{-({\mathrm{e}}^t-1{)}^2}\}}\\ & =& {\displaystyle \frac{{\mathrm{e}}^t}{n}\{\frac{({\mathrm{e}}^{nt}-1)}{({\mathrm{e}}^t-1)}+\frac{(n{\mathrm{e}}^{nt})}{({\mathrm{e}}^t-1)}+\frac{{\mathrm{e}}^t({\mathrm{e}}^{nt}-1)}{-({\mathrm{e}}^t-1{)}^2}\}}\\ & =& {\displaystyle \frac{{\mathrm{e}}^t}{n}\frac{({\mathrm{e}}^{nt}-1)({\mathrm{e}}^t-1)+(n{\mathrm{e}}^{nt})({\mathrm{e}}^t-1)-{\mathrm{e}}^t({\mathrm{e}}^{nt}-1)}{({\mathrm{e}}^t-1{)}^2}}\\ & =& {\displaystyle \frac{{\mathrm{e}}^t}{n}\frac{({\mathrm{e}}^{nt}{\mathrm{e}}^t-{\mathrm{e}}^{nt}-{\mathrm{e}}^t+1)+(n{\mathrm{e}}^{nt}{\mathrm{e}}^t-n{\mathrm{e}}^{nt})-({\mathrm{e}}^{nt}{\mathrm{e}}^t-{\mathrm{e}}^t)}{({\mathrm{e}}^t-1{)}^2}}\\ & =& {\displaystyle \frac{{\mathrm{e}}^t}{n}\frac{{\mathrm{e}}^{nt}{\mathrm{e}}^t-{\mathrm{e}}^{nt}-{\mathrm{e}}^t+1+n{\mathrm{e}}^{nt}{\mathrm{e}}^t-n{\mathrm{e}}^{nt}-{\mathrm{e}}^{nt}{\mathrm{e}}^t+{\mathrm{e}}^t}{({\mathrm{e}}^t-1{)}^2}}\\ & =& {\displaystyle \frac{{\mathrm{e}}^t}{n}\frac{{\mathrm{e}}^{nt}({\mathrm{e}}^t-1+n{\mathrm{e}}^t-n-{\mathrm{e}}^t)+1}{({\mathrm{e}}^t-1{)}^2}}\\ & =& {\displaystyle \frac{{\mathrm{e}}^t}{n}\frac{{\mathrm{e}}^{nt}(n{\mathrm{e}}^t-(n+1))+1}{({\mathrm{e}}^t-1{)}^2}}\\ & =& {\displaystyle \frac{{\mathrm{e}}^t}{n}\frac{n{\mathrm{e}}^{(n+1)t}-(n+1){\mathrm{e}}^{nt}+1}{({\mathrm{e}}^t-1{)}^2}}\end{array}$$

原点周りの一次モーメント=期待値

$$\begin{array}{rcl}{\displaystyle E[X]}& =& {\displaystyle M_X^{(1)}(0)}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left\{\frac{{\mathrm{e}}^t}{n}{\displaystyle \frac{n{\mathrm{e}}^{(n+1)t}-(n+1){\mathrm{e}}^{nt}+1}{({\mathrm{e}}^t-1{)}^2}}\right\}\dots 0\mathrm{を}\mathrm{代}\mathrm{入}\mathrm{す}\mathrm{る}\mathrm{と}\mathrm{分}\mathrm{母}\mathrm{が}0\mathrm{に}\mathrm{な}\mathrm{っ}\mathrm{て}\mathrm{し}\mathrm{ま}\mathrm{う}\mathrm{の}\mathrm{で}\mathrm{極}\mathrm{限}\mathrm{で}\mathrm{考}\mathrm{え}\mathrm{る}．}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left\{\frac{{\mathrm{e}}^t}{n}{\displaystyle \frac{n{\mathrm{e}}^{(n+1)t}-(n+1){\mathrm{e}}^{nt}+1}{{\mathrm{e}}^{2t}-2{\mathrm{e}}^t+1}}\right\}}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left\{\frac{(\frac{t^0}{0!}+\frac{t^1}{1!}+\frac{t^2}{2!}+\frac{t^3}{3!})}{n}\frac{n(\frac{((n+1)t{)}^0}{0!}+\frac{((n+1)t{)}^1}{1!}+\frac{((n+1)t{)}^2}{2!}+\frac{((n+1)t{)}^3}{3!})-(n+1)(\frac{(nt{)}^0}{0!}+\frac{(nt{)}^1}{1!}+\frac{(nt{)}^2}{2!}+\frac{(nt{)}^3}{3!})+1}{(\frac{(2t{)}^0}{0!}+\frac{(2t{)}^1}{1!}+\frac{(2t{)}^2}{2!}+\frac{(2t{)}^3}{3!})-2(\frac{t^0}{0!}+\frac{t^1}{1!}+\frac{t^2}{2!}+\frac{t^3}{3!})+1}\right\}}\\ & & {\displaystyle \dots {\mathrm{e}}^x=\sum _{k=0}^{\mathrm{\infty }}\frac{x^k}{k!}=\frac{x^0}{0!}+\frac{x^1}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots (\mathrm{マ}\mathrm{ク}\mathrm{ロ}\mathrm{ー}\mathrm{リ}\mathrm{ン}\mathrm{展}\mathrm{開}),\mathrm{ひ}\mathrm{と}\mathrm{ま}\mathrm{ず}{t}^3\mathrm{の}\mathrm{項}\mathrm{ま}\mathrm{で}\mathrm{で}\mathrm{計}\mathrm{算}\mathrm{を}\mathrm{進}\mathrm{め}\mathrm{る}．}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left\{\frac{(1+t+\frac{1}{2}{t}^2+\frac{1}{6}{t}^3)}{n}\frac{n(1+(n+1)t+\frac{(n+1{)}^2}{2}{t}^2+\frac{(n+1{)}^3}{6}{t}^3)-(n+1)(1+nt+\frac{n^2}{2}{t}^2+\frac{n^3}{6}{t}^3)+1}{(1+(2t)+\frac{(2t{)}^2}{2}+\frac{(2t{)}^3}{6})-2(1+(t)+\frac{t^2}{2}+\frac{t^3}{6})+1}\right\}}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left\{\frac{(1+t+\frac{1}{2}{t}^2+\frac{1}{6}{t}^3)}{n}\frac{n(1+(n+1)t+\frac{(n+1{)}^2}{2}{t}^2+\frac{(n+1{)}^3}{6}{t}^3)-(n+1)(1+nt+\frac{n^2}{2}{t}^2+\frac{n^3}{6}{t}^3)+1}{1+2t+2t^2+\frac{4}{3}{t}^3-2-2t-t^2-\frac{1}{3}{t}^3+1}\right\}}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left\{\frac{(1+t+\frac{1}{2}{t}^2+\frac{1}{6}{t}^3)}{n}\frac{n(1+(n+1)t+\frac{(n+1{)}^2}{2}{t}^2+\frac{(n+1{)}^3}{6}{t}^3)-(n+1)(1+nt+\frac{n^2}{2}{t}^2+\frac{n^3}{6}{t}^3)+1}{t^2(1+t)}\right\}}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left[\frac{(1+t+\frac{1}{2}{t}^2+\frac{1}{6}{t}^3)}{n{t}^2(1+t)}\{n+n(n+1)t+\frac{n(n+1{)}^2}{2}{t}^2+\frac{n(n+1{)}^3}{6}{t}^3-(n+1)-(n+1)nt-\frac{(n+1)n^2}{2}{t}^2-\frac{(n+1)n^3}{6}{t}^3+1\}\right]}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left[\frac{(1+t+\frac{1}{2}{t}^2+\frac{1}{6}{t}^3)}{n{t}^2(1+t)}\{(n-(n+1)+1)+(n(n+1)-(n+1)n)t+(\frac{n(n+1{)}^2}{2}-\frac{(n+1)n^2}{2})t^2+(\frac{n(n+1{)}^3}{6}-\frac{(n+1)n^3}{6})t^3\}\right]}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left[\frac{(1+t+\frac{1}{2}{t}^2+\frac{1}{6}{t}^3)}{n{t}^2(1+t)}\{(0)+(0)t+(\frac{n+1}{2})n{t}^2+(\frac{(n+1)(2n+1)}{6})n{t}^3\}\right]}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left\{\frac{(1+t+\frac{1}{2}{t}^2+\frac{1}{6}{t}^3)}{n{t}^2(1+t)}n{t}^2(\frac{n+1}{2}+\frac{(n+1)(2n+1)}{6}t)\right\}}\\ & & \dots \mathrm{分}\mathrm{母}\mathrm{は}{t}^2\mathrm{の}\mathrm{項}\mathrm{か}\mathrm{ら}\mathrm{が}\mathrm{残}\mathrm{っ}\mathrm{て}\mathrm{い}\mathrm{る}．t^3\mathrm{以}\mathrm{上}\mathrm{の}\mathrm{項}\mathrm{は}t\mathrm{が}\mathrm{残}\mathrm{る}\mathrm{の}\mathrm{で}\mathrm{マ}\mathrm{ク}\mathrm{ロ}\mathrm{ー}\mathrm{リ}\mathrm{ン}\mathrm{展}\mathrm{開}\mathrm{は}{t}^3\mathrm{で}\mathrm{十}\mathrm{分}\mathrm{と}\mathrm{な}\mathrm{る}．\\ & =& {\displaystyle \underset{t\to 0}{lim}\left\{(1+t+\frac{1}{2}{t}^2+\frac{1}{6}{t}^3)(\frac{n+1}{2}+\frac{(n+1)(2n+1)}{6}t)\right\}}\\ & =& {\displaystyle \frac{n+1}{2}}\\ & & \dots t\mathrm{が}\mathrm{分}\mathrm{子}\mathrm{に}\mathrm{あ}\mathrm{る}(\mathrm{掛}\mathrm{け}\mathrm{ら}\mathrm{れ}\mathrm{て}\mathrm{い}\mathrm{る})\mathrm{項}\mathrm{は}\mathrm{全}\mathrm{て}0．\end{array}$$