確率変数の変数変換 Z=X+Y

確率変数の変数変換 Z=X+Y

Z=X+Y

$$ \begin{eqnarray} p_{X+Y}(z) &=&\int\int \delta(z-(x+y))f(x)g(y)\mathrm{d}x\mathrm{d}y \\&&\;\cdots\;\href{https://shikitenkai.blogspot.com/2020/09/delta-function.html}{\int_{-\infty}^{\infty}\delta(x)f(x)\mathrm{d}x=f(0)} \\&&\;\cdots\;X=x,Y=yの同時確率f(x)g(x)のうちz-(x+y)=0を満たすものだけを足し合わせる. \\&=&\int\int \delta(z-((z-y)+y))f(z-y)g(y)\mathrm{d}y \;\cdots\;z=x+y,x=z-y \\&=&\int \delta(0)f(z-y)g(y)\mathrm{d}y \\&=&\int f(z-y)g(y)\mathrm{d}y \;\cdots\;\delta凾数の引数が全域で0なので，全域でf(z-y)g(y)を足し合わせる積分になる. \end{eqnarray} $$

標準正規分布同士の例

$$ \begin{eqnarray} f_X(x)&=&\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} \;\cdots\;N(0,1)\;(標準\href{https://shikitenkai.blogspot.com/2019/06/binomial-distributionnormal-distribution.html}{正規分布}) \\g_Y(y)&=&\frac{1}{\sqrt{2\pi}}e^{-\frac{y^2}{2}} \;\cdots\;N(0,1)\;(標準\href{https://shikitenkai.blogspot.com/2019/06/binomial-distributionnormal-distribution.html}{正規分布}) \\p_{X+Y}(z)&=&\int\int \delta(z-(x+y))f_X(x)g_Y(y)\mathrm{d}x\mathrm{d}y \\&=&\int\int \delta(z-(x+y))\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}\frac{1}{\sqrt{2\pi}}e^{-\frac{y^2}{2}}\mathrm{d}x\mathrm{d}y \\&=&\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}}e^{-\frac{(z-y)^2}{2}}\frac{1}{\sqrt{2\pi}}e^{-\frac{y^2}{2}} \mathrm{d}y \;\cdots\;z=x+y,x=z-y \\&=&\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{-\frac{(z-y)^2}{2}}e^{-\frac{y^2}{2}} \mathrm{d}y \;\cdots\;\href{https://shikitenkai.blogspot.com/2019/06/continuous-random-variable-expected.html}{\int c f(x) \mathrm{d}x = c \int f(x) \mathrm{d}x} \\&=&\frac{1}{2\pi}\int e^{-\frac{(z-y)^2}{2}-\frac{y^2}{2}} \mathrm{d}y \;\cdots\;A^BA^C=A^{B+C} \\&=&\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{-\frac{(z-y)^2+y^2}{2}} \mathrm{d}y \\&=&\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{-\frac{z^2-2yz+y^2+y^2}{2}}\mathrm{d}y \\&=&\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{-\frac{z^2-2yz+2y^2}{2}}\mathrm{d}y \\&=&\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{-\frac{z^2}{2}+yz-y^2}\mathrm{d}y \\&=&\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{-\frac{z^2}{2}\color{red}{+\frac{z^2}{4}-\frac{z^2}{4}}\color{black}{+yz-y^2}\mathrm{d}y} \\&=&\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{-\frac{z^2}{4}-(\frac{z^2}{4}-yz+y^2)}\mathrm{d}y \\&=&\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{-\frac{z^2}{4}}e^{-(\frac{z^2}{4}-yz+y^2)}\mathrm{d}y \\&=&\frac{1}{2\pi}e^{-\frac{z^2}{4}}\int_{-\infty}^{\infty} e^{-(\frac{z}{2}-y)^2}\mathrm{d}y \\&=&\frac{1}{2\pi}e^{-\frac{z^2}{4}}\int_{\infty}^{-\infty} e^{-t^2}(-1)\mathrm{d}t \\&&\;\cdots\;t=\frac{z}{2}-y,\frac{\mathrm{d}t}{\mathrm{d}y}=-1,\mathrm{d}y=-\mathrm{d}t,\;y:-\infty\rightarrow\infty, t:\infty\rightarrow -\infty \\&=&\frac{1}{2\pi}e^{-\frac{z^2}{4}}(-1)\int_{\infty}^{-\infty} e^{-t^2}\mathrm{d}t \;\cdots\;\href{https://shikitenkai.blogspot.com/2019/06/continuous-random-variable-expected.html}{\int c f(x) \mathrm{d}x = c \int f(x) \mathrm{d}x} \\&=&\frac{1}{2\pi}e^{-\frac{z^2}{4}}(-1)\left(-\int_{-\infty}^{\infty} e^{-t^2}\mathrm{d}t\right) \;\cdots\;\int_{A}^{B} f(x)\mathrm{d}x=-\int_{B}^{A} f(x)\mathrm{d}x \\&=&\frac{1}{2\pi}e^{-\frac{z^2}{4}}\int_{-\infty}^{\infty} e^{-t^2}\mathrm{d}t \\&=&\frac{1}{2\pi}e^{-\frac{z^2}{4}}\sqrt{\pi} \;\cdots\;\href{https://shikitenkai.blogspot.com/2019/06/gaussian-integral.html}{\int_{-\infty}^{\infty} e^{-t^2}\mathrm{d}y=\sqrt{\pi}} \\&=&\frac{1}{2\sqrt{\pi}}e^{-\frac{z^2}{4}} \\&=&\frac{1}{\sqrt{2\pi\cdot2}}e^{-\frac{z^2}{2\cdot2}} \;\cdots\;N(0,2) \end{eqnarray} $$

指数分布同士の例

$$ \begin{eqnarray} f_X(x)&=&\lambda e^{-\lambda x} (x \geq 0, \lambda \gt 0) \;\cdots\;Exp(\lambda)\;(\href{https://shikitenkai.blogspot.com/2020/05/blog-post_0.html}{指数分布}) \\g_Y(y)&=&\lambda e^{-\lambda y} (x \geq 0, \lambda \gt 0) \;\cdots\;Exp(\lambda)\;(\href{https://shikitenkai.blogspot.com/2020/05/blog-post_0.html}{指数分布}) \\p_{X+Y}(z)&=&\int \int \delta(z-(x+y))f_X(x)g_Y(y)\mathrm{d}x\mathrm{d}y \\&=&\int \int \delta(z-(x+y))\lambda e^{-\lambda x} \lambda e^{-\lambda y} \mathrm{d}x\mathrm{d}y \\&=&\int_0^z \lambda e^{-\lambda (z-y)} \lambda e^{-\lambda y}\mathrm{d}y \;\cdots\;z=x+y,x=z-y \\&=&\lambda^2\int_0^z e^{-\lambda (z-y)} e^{-\lambda y}\mathrm{d}y \;\cdots\;\href{https://shikitenkai.blogspot.com/2019/06/continuous-random-variable-expected.html}{\int c f(x) \mathrm{d}x = c \int f(x) \mathrm{d}x} \\&=&\lambda^2\int_0^z e^{-\lambda (z-y)-\lambda y}\mathrm{d}y \;\cdots\;A^BA^C=A^{B+C} \\&=&\lambda^2\int_0^z e^{-\lambda (z-y+y)}\mathrm{d}y \\&=&\lambda^2\int_0^z e^{-\lambda z}\mathrm{d}y \\&=&\lambda^2 e^{-\lambda z}\int_0^z\mathrm{d}y \;\cdots\;\href{https://shikitenkai.blogspot.com/2019/06/continuous-random-variable-expected.html}{\int c f(x) \mathrm{d}x = c \int f(x) \mathrm{d}x} \\&=&\lambda^2 e^{-\lambda z}\left[y\right]_0^z \\&=&\lambda^2 e^{-\lambda z}\left[z-0\right] \\&=&\lambda^2 z e^{-\lambda z} \\&=&\Gamma(2, \lambda) \end{eqnarray} $$ $$ \begin{eqnarray} \Gamma(\alpha, \lambda)&=&\frac{1}{\Gamma(\alpha)}\lambda^{\alpha} x^{\alpha-1} e^{-\lambda x}\;(\href{https://shikitenkai.blogspot.com/2020/05/n.html}{ガンマ分布;\;gamma\;disribution}) \\\Gamma(1, \lambda)&=&\frac{1}{\Gamma(1)}\lambda^1 x^{1-1} e^{-\lambda x}=\frac{1}{1}\lambda e^{-\lambda x}=\lambda e^{-\lambda x} \\&&\;\cdots\;\Gamma(1)=(1-1)!=0!=1 \\&&\;\cdots\;\href{https://shikitenkai.blogspot.com/2020/05/blog-post_0.html}{指数分布} \\\Gamma(2, \lambda)&=&\frac{1}{\Gamma(2)}\lambda^2 x^{2-1} e^{-\lambda x}=\frac{1}{1}\lambda^2xe^{-\lambda x}=\lambda^2xe^{-\lambda x} \\&&\;\cdots\;\Gamma(2)=(2-1)!=1!=1 \\&&\;\cdots\;指数分布に従う確率変数同士の和の分布,\;\alpha=2のガンマ分布 \end{eqnarray} $$

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)． $$ \begin{eqnarray} \frac{\mathrm{e}^{-\beta t}}{\Gamma(\alpha)}\beta^\alpha t^\left(\alpha-1\right) &=&\frac{\mathrm{e}^{-\beta t}}{\Gamma(1)}\beta^1 t^\left(1-1\right)\;\cdots\;\alpha=1\\ &=&\frac{\mathrm{e}^{-\beta t}}{1}\beta t^\left(0\right)\;\cdots\;\Gamma(n)=(n-1)!,\Gamma(1)=(1-1)!=0!=1\\ &=&\mathrm{e}^{-\beta t}\beta\;\cdots\;t^0=1\\ &=&\href{https://shikitenkai.blogspot.com/2020/05/blog-post_0.html}{\lambda\mathrm{e}^{-\lambda t}}\;\cdots\;\beta=\lambda\\ \end{eqnarray} $$