連続型確率変数(continuous 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

$$\begin{array}{rcl}{\displaystyle M_X^{(1)}}& =& {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\left\{{\displaystyle \frac{{\mathrm{e}}^{tb}-{\mathrm{e}}^{ta}}{t(b-a)}{\displaystyle }}\right\}}\\ & =& {\displaystyle \frac{1}{b-a}{\displaystyle \left\{{\displaystyle {\left(t^{-1}\right)}^{\prime }({\mathrm{e}}^{tb}-{\mathrm{e}}^{ta}){\displaystyle +\left(t^{-1}\right){({\mathrm{e}}^{tb}-{\mathrm{e}}^{ta})}^{\prime }{\displaystyle }}}\right\}}}\\ & =& {\displaystyle \frac{1}{b-a}{\displaystyle \left\{{\displaystyle (-t^{-2})({\mathrm{e}}^{tb}-{\mathrm{e}}^{ta}){\displaystyle +\left(t^{-1}\right)(b{\mathrm{e}}^{tb}-a{\mathrm{e}}^{ta}){\displaystyle }}}\right\}}}\\ & =& {\displaystyle \frac{1}{b-a}{\displaystyle \left({\displaystyle -\frac{{\mathrm{e}}^{tb}-{\mathrm{e}}^{ta}}{t^2}{\displaystyle +\frac{b{\mathrm{e}}^{tb}-a{\mathrm{e}}^{ta}}{t}{\displaystyle }}}\right)}}\\ & =& {\displaystyle \frac{1}{b-a}{\displaystyle \left\{{\displaystyle -\frac{{\mathrm{e}}^{tb}-{\mathrm{e}}^{ta}}{t^2}{\displaystyle +\frac{(b{\mathrm{e}}^{tb}-a{\mathrm{e}}^{ta})t}{t^2}{\displaystyle }}}\right\}}}\\ & =& {\displaystyle \frac{1}{b-a}{\displaystyle \left({\displaystyle \frac{tb{\mathrm{e}}^{tb}-{\mathrm{e}}^{tb}-ta{\mathrm{e}}^{ta}+{\mathrm{e}}^{ta}}{t^2}{\displaystyle }}\right)}}\\ & =& {\displaystyle \frac{1}{t^2(b-a)}{\displaystyle \left\{{\displaystyle (tb-1){\mathrm{e}}^{tb}-(ta-1){\mathrm{e}}^{ta}{\displaystyle }}\right\}}}\\ & =& {\displaystyle \frac{(tb-1){\mathrm{e}}^{tb}-(ta-1){\mathrm{e}}^{ta}}{t^2(b-a)}}\end{array}$$

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\{{\displaystyle \frac{(tb-1){\mathrm{e}}^{tb}-(ta-1){\mathrm{e}}^{ta}}{t^2(b-a)}{\displaystyle }}\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[{\displaystyle \frac{1}{t^2(b-a)}\left\{{\displaystyle (tb-1){\mathrm{e}}^{tb}-(ta-1){\mathrm{e}}^{ta}{\displaystyle }}\right\}{\displaystyle }}\right]}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left[{\displaystyle \frac{1}{t^2(b-a)}\left\{{\displaystyle (tb-1)(\frac{(tb{)}^0}{0!}+\frac{(tb{)}^1}{1!}+\frac{(tb{)}^2}{2!}){\displaystyle -(ta-1)(\frac{(ta{)}^0}{0!}+\frac{(ta{)}^1}{1!}+\frac{(ta{)}^2}{2!}){\displaystyle }}}\right\}{\displaystyle }}\right]}\\ & & \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!}+\cdots (\mathrm{マ}\mathrm{ク}\mathrm{ロ}\mathrm{ー}\mathrm{リ}\mathrm{ン}\mathrm{展}\mathrm{開}),t^2\mathrm{が}\mathrm{分}\mathrm{母}\mathrm{に}\mathrm{あ}\mathrm{る}\mathrm{の}\mathrm{で}{t}^3\mathrm{の}\mathrm{項}\mathrm{以}\mathrm{上}\mathrm{は}\mathrm{分}\mathrm{子}\mathrm{に}t\mathrm{が}\mathrm{残}\mathrm{る}\mathrm{こ}\mathrm{と}\mathrm{に}\mathrm{な}\mathrm{る}\mathrm{の}\mathrm{で}{t}^2\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[{\displaystyle \frac{1}{t^2(b-a)}\left\{{\displaystyle (tb-1)(1+tb+t^2\frac{b^2}{2}){\displaystyle -(ta-1)(1+ta+t^2\frac{a^2}{2}){\displaystyle }}}\right\}{\displaystyle }}\right]}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left[{\displaystyle \frac{1}{t^2(b-a)}\left[{\displaystyle \{(tb+t^2{b}^2+t^3\frac{b^3}{2})-(1+tb+t^2\frac{b^2}{2})\}{\displaystyle -\{(ta+t^2{a}^2+t^3\frac{a^3}{2})-(1+ta+t^2\frac{a^2}{2})\}{\displaystyle }}}\right]{\displaystyle }}\right]}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left[{\displaystyle \frac{1}{t^2(b-a)}\left\{{\displaystyle (-1+t(b-b)+t^2(b^2-\frac{b^2}{2})+t^3\frac{b^3}{2}){\displaystyle -(-1+t(a-a)+t^2(a^2-\frac{a^2}{2})+t^3\frac{a^3}{2}){\displaystyle }}}\right\}{\displaystyle }}\right]}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left[{\displaystyle \frac{1}{t^2(b-a)}\left\{{\displaystyle (-1+t^2\frac{b^2}{2}+t^3\frac{b^3}{2}){\displaystyle -(-1+t^2\frac{a^2}{2}+t^3\frac{a^3}{2}){\displaystyle }}}\right\}{\displaystyle }}\right]}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left\{{\displaystyle \frac{1}{t^2(b-a)}\left({\displaystyle t^2\frac{b^2-a^2}{2}{\displaystyle +t^3\frac{b^3-a^3}{2}{\displaystyle }}}\right){\displaystyle }}\right\}}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left[\frac{1}{t^2(b-a)}\left\{{\displaystyle t^2\frac{(b-a)(b+a)}{2}{\displaystyle +t^3\frac{(b-a)(b^2+ab+a^2)}{2}{\displaystyle }}}\right\}\right]\dots b^2-a^2=(b-a)(b+a),b^3-a^3=(b-a)(b^2+ab+a^2)}\\ & =& {\displaystyle \underset{t\to 0}{lim}{\displaystyle \left\{{\displaystyle \frac{(b+a)}{2}{\displaystyle +t\frac{(b^2+ab+a^2)}{2}{\displaystyle }}}\right\}}}\\ & =& {\displaystyle \frac{b+a}{2}=\frac{a+b}{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}$$