連続型確率変数(continuous random variable) の一様分布(uniform distribution)の分散(variance)

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

積率母凾数の二階微分

$$\begin{array}{rcl}{\displaystyle M_X^{(2)}}& =& {\displaystyle \frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}\left\{{\displaystyle \frac{{\mathrm{e}}^{tb}-{\mathrm{e}}^{ta}}{t(b-a)}{\displaystyle }}\right\}}\\ & =& {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\left\{{\displaystyle \frac{(tb-1){\mathrm{e}}^{tb}-(ta-1){\mathrm{e}}^{ta}}{t^2(b-a)}{\displaystyle }}\right\}}\\ & =& {\displaystyle \frac{1}{b-a}\left[{\displaystyle (t^{-2}{)}^{\prime }\{(tb-1){\mathrm{e}}^{tb}-(ta-1){\mathrm{e}}^{ta}\}{\displaystyle +(t^{-2}){\{(tb-1){\mathrm{e}}^{tb}-(ta-1){\mathrm{e}}^{ta}\}}^{\prime }{\displaystyle }}}\right]}\\ & =& {\displaystyle \frac{1}{b-a}\left[{\displaystyle (-2t^{-3})\{(tb-1){\mathrm{e}}^{tb}-(ta-1){\mathrm{e}}^{ta}\}{\displaystyle +(t^{-2})[{\{(tb-1){\mathrm{e}}^{tb}\}}^{\prime }-{\{(ta-1){\mathrm{e}}^{ta}\}}^{\prime }]{\displaystyle }}}\right]}\\ & =& {\displaystyle \frac{1}{b-a}\left[{\displaystyle (-2t^{-3})\{(tb-1){\mathrm{e}}^{tb}-(ta-1){\mathrm{e}}^{ta}\}{\displaystyle +(t^{-2})[\{(tb-1{)}^{\prime }{\mathrm{e}}^{tb}+(tb-1)({\mathrm{e}}^{tb}{)}^{\prime }\}-\{(ta-1{)}^{\prime }{\mathrm{e}}^{ta}+(ta-1)({\mathrm{e}}^{ta}{)}^{\prime }\}]{\displaystyle }}}\right]}\\ & =& {\displaystyle \frac{1}{b-a}\left[{\displaystyle (-2t^{-3})\{(tb-1){\mathrm{e}}^{tb}-(ta-1){\mathrm{e}}^{ta}\}{\displaystyle +(t^{-2})[\{b{\mathrm{e}}^{tb}+(tb-1)b{\mathrm{e}}^{tb}\}-\{a{\mathrm{e}}^{ta}+(ta-1)a{\mathrm{e}}^{ta}\}]{\displaystyle }}}\right]}\\ & =& {\displaystyle \frac{1}{b-a}\left[\frac{{\displaystyle -2\{(tb-1){\mathrm{e}}^{tb}-(ta-1){\mathrm{e}}^{ta}\}{\displaystyle +t[\{b{\mathrm{e}}^{tb}+(tb-1)b{\mathrm{e}}^{tb}\}-\{a{\mathrm{e}}^{ta}+(ta-1)a{\mathrm{e}}^{ta}\}]}}}{t^3}{\displaystyle }\right]}\\ & =& {\displaystyle \frac{1}{t^3(b-a)}\left[{\displaystyle -2(tb-1){\mathrm{e}}^{tb}+2(ta-1){\mathrm{e}}^{ta}{\displaystyle +tb{\mathrm{e}}^{tb}+tb(tb-1){\mathrm{e}}^{tb}{\displaystyle -ta{\mathrm{e}}^{ta}-ta(ta-1){\mathrm{e}}^{ta}{\displaystyle }}}}\right]}\\ & =& {\displaystyle \frac{1}{t^3(b-a)}\left[{\displaystyle \{-2(tb-1)+tb+tb(tb-1)\}{\mathrm{e}}^{tb}{\displaystyle +\{2(ta-1)-ta-ta(ta-1)\}{\mathrm{e}}^{ta}{\displaystyle }}}\right]}\\ & =& {\displaystyle \frac{1}{t^3(b-a)}\left\{{\displaystyle ((tb-1{)}^2+1){\mathrm{e}}^{tb}{\displaystyle -((ta-1{)}^2+1){\mathrm{e}}^{ta}{\displaystyle }}}\right\}}\\ & =& {\displaystyle \frac{((tb-1{)}^2+1){\mathrm{e}}^{tb}-((ta-1{)}^2+1){\mathrm{e}}^{ta}}{t^3(b-a)}}\end{array}$$

原点周りの二次モーメント

$$\begin{array}{rcl}{\displaystyle E[X^2]}& =& {\displaystyle M_X^{(2)}(0)}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left\{{\displaystyle \frac{((tb-1{)}^2+1){\mathrm{e}}^{tb}-((ta-1{)}^2+1){\mathrm{e}}^{ta}}{t^3(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^3(b-a)}{\displaystyle \left\{{\displaystyle ((tb-1{)}^2+1){\mathrm{e}}^{tb}-((ta-1{)}^2+1){\mathrm{e}}^{ta}{\displaystyle }}\right\}{\displaystyle }}}\right]}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left[{\displaystyle \frac{1}{t^3(b-a)}{\displaystyle \left\{{\displaystyle ((tb-1{)}^2+1)(\frac{(tb{)}^0}{0!}+\frac{(tb{)}^1}{1!}+\frac{(tb{)}^2}{2!}+\frac{(tb{)}^3}{3!}){\displaystyle -((ta-1{)}^2+1)(\frac{(ta{)}^0}{0!}+\frac{(ta{)}^1}{1!}+\frac{(ta{)}^2}{2!}+\frac{(ta{)}^3}{3!}){\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!}+\frac{x^3}{3!}+\cdots (\mathrm{マ}\mathrm{ク}\mathrm{ロ}\mathrm{ー}\mathrm{リ}\mathrm{ン}\mathrm{展}\mathrm{開}),t^3\mathrm{が}\mathrm{分}\mathrm{母}\mathrm{に}\mathrm{あ}\mathrm{る}\mathrm{の}\mathrm{で}{t}^4\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}^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[{\displaystyle \frac{1}{t^3(b-a)}{\displaystyle \left\{{\displaystyle (t^2{b}^2-2tb+2)(1+tb+t^2\frac{b^2}{2}+t^3\frac{b^3}{6}){\displaystyle -(t^2{a}^2-2ta+2)(1+ta+t^2\frac{a^2}{2}+t^3\frac{a^3}{6}){\displaystyle }}}\right\}{\displaystyle }}}\right]}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left[{\displaystyle \frac{1}{t^3(b-a)}{\displaystyle [\{(t^2{b}^2+t^3{b}^3+t^4\frac{b^4}{2}+t^5\frac{b^5}{6})+(-2tb-2t^2{b}^2-t^3{b}^3-t^4\frac{b^4}{3})+(2+2tb+t^2{b}^2+t^3\frac{b^3}{3})\}-\{(t^2{a}^2+t^3{a}^3+t^4\frac{a^4}{2}+t^5\frac{a^5}{6})+(-2ta-2t^2{a}^2-t^3{a}^3-t^4\frac{a^4}{3})+(2+2ta+t^2{a}^2+t^3\frac{a^3}{3})\}{\displaystyle }]{\displaystyle }}}\right]}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left[{\displaystyle \frac{1}{t^3(b-a)}{\displaystyle \{(2+t^3\frac{b^3}{3}+t^4\frac{b^4}{6}+t^5\frac{b^5}{6})-(2+t^3\frac{a^3}{3}+t^4\frac{a^4}{6}+t^5\frac{a^5}{6}){\displaystyle }\}{\displaystyle }}}\right]}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left[{\displaystyle \frac{1}{t^3(b-a)}{\displaystyle \{t^3\frac{b^3-a^3}{3}+t^4\frac{b^4-a^4}{6}+t^5\frac{b^5-a^5}{6}{\displaystyle }\}{\displaystyle }}}\right]}\\ & =& {\displaystyle \underset{t\to 0}{lim}\left[{\displaystyle \frac{1}{t^3(b-a)}{\displaystyle \{t^3\frac{(b-a)(b^2+ab+a^2)}{3}+t^4\frac{(b-a)(b+a)(a^2+b^2)}{6}+t^5\frac{(b-a)\frac{b^5-a^5}{b-a}}{6}{\displaystyle }\}{\displaystyle }}}\right]}\\ & =& {\displaystyle \underset{t\to 0}{lim}\{\frac{b^2+ab+a^2}{3}+t\frac{(b+a)(a^2+b^2)}{6}+t^2\frac{\left(\frac{b^5-a^5}{b-a}\right)}{6}{\displaystyle }\}}\\ & =& {\displaystyle \frac{b^2+ab+a^2}{3}=\frac{a^2+ab+b^2}{3}\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}$$

分散(二次の中心モーメント)

$$\begin{array}{rcl}{\displaystyle V[X]}& =& {\displaystyle E[X^2]-E[X{]}^2}\\ & =& {\displaystyle \frac{b^2+ab+a^2}{3}-{\left(\frac{b+a}{2}\right)}^2}\\ & =& {\displaystyle \frac{b^2+ab+a^2}{3}-\frac{b^2+2ab+a^2}{4}}\\ & =& {\displaystyle \frac{4(b^2+ab+a^2)-3(b^2+2ab+a^2)}{12}}\\ & =& {\displaystyle \frac{4b^2+4ab+4a^2-3b^2-6ab-3a^2}{12}}\\ & =& {\displaystyle \frac{b^2-2ab+a^2}{12}}\\ & =& {\displaystyle \frac{(b-a{)}^2}{12}}\end{array}$$