離散型確率変数(discrete random variable) / 分散(variance)
$$\begin{array}{rcl}
母集団(population)の確率変数&:&X,Y\\
確率質量凾数(probability\,mass\,function,\,PMF)&:&f_X(x)=P(X=x_i)\\
&&\displaystyle\sum_{i=1}^{\infty} f_X(x_i)=1\\
累積分布凾数(cumulative\,distribution\,function,\,CDF)&:&\displaystyle F_X(x)=\sum_{i:x_i \leq x} f_X(x_i)\\
同時確率質量凾数(joint\,probability\,mass\,function)&:&f_X(x, y)=P(X=x_i,\,Y=y_j)\\
&&\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} f_{XY}(x_i, y_j)=1\\
周辺確率質量凾数(marginal\, probability\, mass\, function)&:&\displaystyle f_X(x)=\sum_{j=1}^{\infty} f_{XY}(x, y_i)\\
\end{array}$$
$$\begin{array}{rcl}
V[X]&\equiv&E[(X-E[X])^2]\\
&=&\displaystyle\sum_{i=1}^{\infty} (x_i-E[X])^2 f_X(x_i) \\
&=&\displaystyle\sum_{i=1}^{\infty} (x_i^2-2E[X]x_i+E[X]^2) f_X(x_i) \\
&=&\displaystyle\sum_{i=1}^{\infty} (x_i^2f_X(x_i)-2E[X]x_if_X(x_i)+E[X]^2f_X(x_i)) \\
&=&\displaystyle\sum_{i=1}^{\infty} (x_i^2) f_X(x_i)
-2E[X]\displaystyle\sum_{i=1}^{\infty} (x_i) f_X(x_i)
+E[X]^2\displaystyle\sum_{i=1}^{\infty} f_X(x_i)\\
&=&E[X^2] -2E[X]E[X] +E[X]^2 1\\
&=&E[X^2] -2E[X]^2 +E[X]^2\\
&=&E[X^2]-E[X]^2\\
V[cX] &=&E[(cX-E[cX])^2]\,\dotso\,cは定数\\
&=&\displaystyle\sum_{i=1}^{\infty} (c x_i - E[c X])^2 f_X(x_i) \\
&=&\displaystyle\sum_{i=1}^{\infty} (c x_i - c E[X])^2 f_X(x_i) \\
&=&\displaystyle\sum_{i=1}^{\infty} (c (x_i - E[X]))^2 f_X(x_i) \\
&=&\displaystyle\sum_{i=1}^{\infty} c^2 (x_i-E[X])^2 f_X(x_i) \\
&=&c^2\displaystyle\sum_{i=1}^{\infty} (x_i-E[X])^2 f_X(x_i) \\
&=&c^2 V[X] \\
V[X \pm t]&=&E[((X \pm t)-E[X \pm t])^2]\,\dotso\,tは定数\\
&=&\displaystyle\sum_{i=1}^{\infty} ((x_i \pm t)-E[X \pm t])^2 f_X(x_i) \\
&=&\displaystyle\sum_{i=1}^{\infty} ((x_i \pm t)-(E[X] \pm t))^2 f_X(x_i) \\
&=&\displaystyle\sum_{i=1}^{\infty} (x_i \pm t - E[X] \mp t))^2 f_X(x_i) \\
&=&\displaystyle\sum_{i=1}^{\infty} (x_i-E[X])^2 f_X(x_i) \\
&=&V[X] \\
V[X \pm Y]&=&E[((X \pm Y)-E[X \pm Y])^2]\\
&=&\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} ((x_i \pm y_j)-E[X \pm Y])^2 f_{XY}(x_i, y_j) \\
&=&\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} ((x_i \pm y_j)-(E[X] \pm E[Y]))^2 f_{XY}(x_i, y_j) \\
&=&\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} ((x_i - E[X]) \pm (y_j - E[Y]))^2 f_{XY}(x_i, y_j) \\
&=&\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} ((x_i - E[X])^2 \pm 2(x_i - E[X])(y_j - E[Y]) +(y_j - E[Y])^2) f_{XY}(x_i, y_j) \\
&=&\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} ((x_i - E[X])^2 f_{XY}(x_i, y_j) \pm 2(x_i - E[X])(y_j - E[Y]) f_{XY}(x_i, y_j) +(y_j - E[Y])^2 f_{XY}(x_i, y_j)) \\
&=&\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} (x_i - E[X])^2 f_{XY}(x_i, y_j)
\pm 2\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}(x_i - E[X])(y_j - E[Y]) f_{XY}(x_i, y_j)
+ \displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} (y_j - E[Y])^2 f_{XY}(x_i, y_j) \\
&=&\displaystyle\sum_{i=1}^{\infty}(x_i - E[X])^2\sum_{j=1}^{\infty}f_{XY}(x_i, y_j)
\pm 2\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}(x_i - E[X])(y_j - E[Y]) f_{XY}(x_i, y_j)
+ \displaystyle\sum_{j=1}^{\infty} (y_j - E[Y])^2 \sum_{i=1}^{\infty} f_{XY}(x_i, y_j) \\
&=&\displaystyle\sum_{i=1}^{\infty} (x_i - E[X])^2 f_{X}(x_i)
\pm 2\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}(x_i - E[X])(y_j - E[Y]) f_{XY}(x_i, y_j)
+ \displaystyle\sum_{j=1}^{\infty} (y_j - E[Y])^2 f_{Y}(y_j)\,\dotso\,周辺確率質量凾数を適用 \\
&=&V[X] \pm 2Cov[X,Y] +V[Y]\,\dotso\,Cov[X,Y]=\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}(x_i - E[X])(y_j - E[Y]) f_{XY}(x_i, y_j)\\
\end{array}$$
離散型確率変数(discrete random variable) / 期待値(expected value)
$$\begin{array}{rcl}
母集団(population)の確率変数&:&X,Y\\
確率質量凾数(probability\,mass\,function,\,PMF)&:&f_X(x)=P(X=x_i)\\
&&\displaystyle\sum_{i=1}^{\infty} f_X(x_i)=1\\
累積分布凾数(cumulative\,distribution\,function,\,CDF)&:&\displaystyle F_X(x)=\sum_{i:x_i \leq x} f_X(x_i)\\
同時確率質量凾数(joint\,probability\,mass\,function)&:&f_X(x, y)=P(X=x_i,\,Y=y_j)\\
&&\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} f_{XY}(x_i, y_j)=1\\
周辺確率質量凾数(marginal\, probability\, mass\, function)&:&\displaystyle f_X(x)=\sum_{j=1}^{\infty} f_{XY}(x, y_i)\\
\end{array}$$
$$\begin{array}{rcl}
E[g(X)]&\equiv&\displaystyle\sum_{i=1}^{\infty} g(x_i) f_X(x_i) \\
E[X]&=&\displaystyle\sum_{i=1}^{\infty} (x_i) f_X(x_i) \\
E[cX] &=&\displaystyle\sum_{i=1}^{\infty} (c x_i) f_X(x_i) \,\dotso\,cは定数\\
&=&c\displaystyle\sum_{i=1}^{\infty} x_i f_X(x_i) \\
&=&c E[X] \\
E[X \pm t]&=&\displaystyle\sum_{i=1}^{\infty} (x_i \pm t) f_X(x_i) \,\dotso\,tは定数\\
&=&\displaystyle\sum_{i=1}^{\infty} (x_i f_X(x_i) \pm t f_X(x_i)) \\
&=&\displaystyle\sum_{i=1}^{\infty} x_i f_X(x_i)
\pm \displaystyle\sum_{i=1}^{\infty} t f_X(x_i) \\
&=&\displaystyle\sum_{i=1}^{\infty} x_i f_X(x_i)
\pm t \displaystyle\sum_{i=1}^{\infty} f_X(x_i) \\
&=&E[X] \pm t\\
E[X \pm Y]&=&\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} (x_i \pm y_j) f_{XY}(x_i, y_j) \\
&=&\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} (x_i f_{XY}(x_i, y_j) \pm y_j f_{XY}(x_i, y_j)) \\
&=&\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} x_i f_{XY}(x_i, y_j)
\pm \displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} y_j f_{XY}(x_i, y_j)\\
&=&\displaystyle\sum_{i=1}^{\infty} x_i \sum_{j=1}^{\infty} f_{XY}(x_i, y_j)
\pm \displaystyle\sum_{j=1}^{\infty} y_j \sum_{i=1}^{\infty}f_{XY}(x_i, y_j)\\
&=&\displaystyle\sum_{i=1}^{\infty} x_i f_{X}(x_i)
\pm \displaystyle\sum_{j=1}^{\infty} y_j f_{Y}(y_j)\,\dotso\,周辺確率質量凾数を適用\\
&=& E[X] \pm E[Y] \\
\end{array}$$
X,Yが独立の場合
$$\begin{array}{rcl} E[XY]&=&\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}(x_i y_j)f_{XY}(x_i, y_j)\\ &=&\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}x_iy_jf_{X}(x_i)f_{Y}(y_i) \,\dotso\,X,Yが独立\,f_{XY}(x, y)=f_{X}(x)f_{Y}(y)\\ &=&\displaystyle\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}x_if_{X}(x_i)y_jf_{Y}(y_i)\\ &=&\displaystyle\left(\sum_{i=1}^{\infty}x_if_{X}(x_i)\right)\left(\sum_{j=1}^{\infty} y_jf_{Y}(y_i)\right) \,\dotso\,\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}a_ib_j=\sum_{i=1}^{\infty}a_i\sum_{j=1}^{\infty}b_j\\ &=&E[X]E[Y] \end{array}$$ガウス積分(Gaussian integral)
$$\begin{array}{rcl}
\displaystyle \int_{\mathbb{R}}\mathrm{e}^{-(x^2+y^2)}\mathrm{d}A
&=& \displaystyle \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\mathrm{e}^{-(x^2+y^2)}\mathrm{d}x\mathrm{d}y\dotso \mathrm{d}A=\mathrm{d}x\mathrm{d}y\\
&=& \displaystyle \int_{-\infty}^{\infty}\mathrm{e}^{-x^2}\mathrm{d}x\int_{-\infty}^{\infty}\mathrm{e}^{-y^2}\mathrm{d}y\\
&=& \displaystyle \left(\int_{-\infty}^{\infty}\mathrm{e}^{-t^2}\mathrm{d}t\right)^2\\
\displaystyle \int_{\mathbb{R}}\mathrm{e}^{-(x^2+y^2)}\mathrm{d}A
&=& \displaystyle \int_{0}^{2\pi}\int_{0}^{\infty}\mathrm{e}^{-r^2}r\mathrm{d}r\mathrm{d}\theta\dotso \mathrm{d}A=r\mathrm{d}r\mathrm{d}\theta\\
&=& \displaystyle 2\pi\int_{0}^{\infty}\mathrm{e}^{-r^2}r\mathrm{d}r\\
&=& \displaystyle 2\pi\int_{0}^{\infty}\mathrm{e}^{-z} r\frac{1}{2r}\mathrm{d}z
\dotso z=r^2,\frac{\mathrm{d}z}{\mathrm{d}r}=2r,\mathrm{d}r=\frac{1}{2r}\mathrm{d}z\\
&=& \displaystyle \pi\int_{0}^{\infty}\mathrm{e}^{-z}\mathrm{d}z\\
&=& \displaystyle \pi\left[-\mathrm{e}^{-z}\right]_0^{\infty}
= \displaystyle \pi\left[\left(-\mathrm{e}^{-\infty}\right)-\left(-\mathrm{e}^{-0}\right)\right]
= \pi\left[0-\left(-1\right)\right]\\
&=& \displaystyle \pi\\
\displaystyle \left(\int_{-\infty}^{\infty}\mathrm{e}^{-t^2}\mathrm{d}t\right)^2
&=& \displaystyle \pi\\
\displaystyle \int_{-\infty}^{\infty}\mathrm{e}^{-t^2}\mathrm{d}t
&=& \displaystyle \sqrt{\pi}
\end{array}$$
\(-t^2\)でなく\(-\left(\frac{x+b}{c}\right)^2\)の場合(\(b,cは定数\))
$$\begin{array}{rcl} \displaystyle \int_{-\infty}^{\infty}\mathrm{e}^{-\left(\frac{x+b}{c}\right)^2}\mathrm{d}x &=& \displaystyle \int_{-\infty}^{\infty}\mathrm{e}^{-z^2}\,c\,\mathrm{d}z \,\dotso\,\frac{x+b}{c}=z,\frac{\mathrm{d}z}{\mathrm{d}x}=\frac{1}{c},\mathrm{d}x=c\mathrm{d}z\\ &=& \displaystyle c\int_{-\infty}^{\infty}\mathrm{e}^{-z^2}\mathrm{d}z\\ &=& \displaystyle c\sqrt{\pi}\\ \end{array}$$
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