正規分布の差の再現性
分散，データ数，ともに異なる場合
独立な確率変数X，Yがそれぞれ確率分布PX(x), PY(y)に従うとします．
各確率変数の和，X+Yが従う確率分布をPX-Y(z)とする 確率P(X-Y=z)を考えると，
X-Y=z
となるのは，
X=x, Y=z-x
としたとき，両者の差がｚとなるすべての組み合わせとなります．
XはPX(x)，YはPY(z-x)に従うので，両者が同時に起こる確率は，PX(x)PY(z-x) です．
\( \Large P_{X\color{red}{-}Y} (z) = \displaystyle \int_{-\infty}^{ \infty } P_{X,Y} (t,\color{red}{t-z}) dt \)
X，Yがそれぞれ独立であると仮定したので，\( \Large P_{X,Y} (z) = P_X(x) P_Y (y) \) が成立します．したがって，
\( \Large P_{X\color{red}{-}Y} (z) = \displaystyle \int_{-\infty}^{ \infty } P_{X} (t) P_Y(z-t) dt \)
となります．それぞれが正規分布に従うので，
\( \Large P_{X} (x) = \frac{1}{\sqrt{2 \pi \frac{\sigma_X^2}{n_X}}} exp \left[- \frac{(x-\mu_X)^2}{2 \frac{\sigma_X^2}{n_X}} \right] \)
\( \Large P_{Y} (x) = \frac{1}{\sqrt{2 \pi \frac{\sigma_Y^2}{n_Y}}} exp \left[- \frac{(x-\mu_Y)^2}{2 \frac{\sigma_Y^2}{n_Y}} \right] \)
となりますので，代入すると，
\( \Large \begin{eqnarray} P_{X\color{red}{-}Y} (z) &=& \displaystyle \int_{-\infty}^{ \infty } \frac{1}{\sqrt{2 \pi \frac{\sigma_X^2}{n_X}}}
exp \left[- \frac{(t-\mu_X)^2}{2 \frac{\sigma_X^2}{n_X}} \right]
\frac{1}{\sqrt{2 \pi \frac{\sigma_Y^2}{n_Y}}}
exp \left[- \frac{(\color{red}{t-z}-\mu_Y)^2}{2 \frac{\sigma_Y^2}{n_Y}} \right]dt
\\
&=& \frac{1}{2 \pi \sqrt{\frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}}} \displaystyle \int_{-\infty}^{ \infty }
exp \left[- \frac{(t-\mu_X)^2}{2 \frac{\sigma_X^2}{n_X}} - \frac{(\color{red}{t-z}-\mu_Y)^2}{2 \frac{\sigma_Y^2}{n_Y}} \right]dt
\\
\end{eqnarray} \)
と表すことができます．
指数部分は，
\( \begin{eqnarray} && - \frac{(t-\mu_X)^2}{2 \frac{\sigma_X^2}{n_X}} - \frac{(\color{red}{t-z}-\mu_Y)^2}{2 \frac{\sigma_Y^2}{n_Y}} \\
&=&
- \frac{\frac{\sigma_Y^2}{n_Y}(t^2-2 \mu_X t+ \mu_X^2) + \frac{\sigma_X^2}{n_X} (z^2+t^2+\mu_Y^2 \color{red}{-} 2 \mu_Y t-2zt \color{red}{+} 2 \mu_Y z)}{2 \frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}} \\
&=&
- \frac{(\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}) \color{red}{t^2}-2(\frac{\sigma_Y^2}{n_Y} \mu_X + \frac{\sigma_X^2}{n_X} z \color{red}{+} \frac{\sigma_X^2}{n_X} \mu_Y) \color{red}{t} + \frac{\sigma_Y^2}{n_Y} \mu_X^2 + \frac{\sigma_X^2}{n_X} z^2 + \frac{\sigma_X^2}{n_X} \mu_Y^2 \color{red}{+} 2 \frac{\sigma_X^2}{n_X} \mu_Y z} {2 \frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}}\\
&=&
- \frac{\frac{\sigma_X^2}{n_X}+\frac{\sigma_Y^2}{n_Y}}{2\frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}} \left\{ \color{red}{t^2}-\frac{2(\frac{\sigma_Y^2}{n_Y} \mu_X + \frac{\sigma_X^2}{n_X} z \color{red}{+} \frac{\sigma_X^2}{n_X} \mu_Y)}{\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}} \color{red}{t} + \frac{
\frac{\sigma_Y^2}{n_Y} \mu_X^2 + \frac{\sigma_X^2}{n_X} z^2 + \frac{\sigma_X^2}{n_X} \mu_Y^2 \color{red}{+} 2 \frac{\sigma_X^2}{n_X} \mu_Y z} { \frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}} \right\} \\
\end{eqnarray} \)
\( \Large t^2 + 2b t + c = (t+b)^2 - b^2 +c \)より，
\( \begin{eqnarray}
&=&
- \frac{\frac{\sigma_X^2}{n_X}+\frac{\sigma_Y^2}{n_Y}}{2\frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}}
\left\{ \color{red}{t^2}-\frac{2(\frac{\sigma_Y^2}{n_Y} \mu_X + \frac{\sigma_X^2}{n_X} z \color{red}{+} \frac{\sigma_X^2}{n_X} \mu_Y)}{\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}} \color{red}{t} + \left[ \frac{(\frac{\sigma_Y^2}{n_Y} \mu_X + \frac{\sigma_X^2}{n_X} z \color{red}{+} \frac{\sigma_X^2}{n_X} \mu_Y)}{\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}} \right]^2 - \left[ \frac{(\frac{\sigma_Y^2}{n_Y} \mu_X + \frac{\sigma_X^2}{n_X} z \color{red}{+} \frac{\sigma_X^2}{n_X} \mu_Y)}{\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}} \right]^2+ \frac{
\frac{\sigma_Y^2}{n_Y} \mu_X^2 + \frac{\sigma_X^2}{n_X} z^2 + \frac{\sigma_X^2}{n_X} \mu_Y^2 \color{red}{+} 2 \frac{\sigma_X^2}{n_X} \mu_Y z} { \frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}} \right\} \\
&=&
- \frac{\frac{\sigma_X^2}{n_X}+\frac{\sigma_Y^2}{n_Y}}{2\frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}}
\left\{ \left[ \color{red}{t}-\frac{(\frac{\sigma_Y^2}{n_Y} \mu_X + \frac{\sigma_X^2}{n_X} z \color{red}{+} \frac{\sigma_X^2}{n_X} \mu_Y)}{\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}} \right]^2 - \left[ \frac{(\frac{\sigma_Y^2}{n_Y} \mu_X + \frac{\sigma_X^2}{n_X} z \color{red}{+} \frac{\sigma_X^2}{n_X} \mu_Y)}{\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}} \right]^2+ \frac{
\frac{\sigma_Y^2}{n_Y} \mu_X^2 + \frac{\sigma_X^2}{n_X} z^2 + \frac{\sigma_X^2}{n_X} \mu_Y^2 \color{red}{+} 2 \frac{\sigma_X^2}{n_X} \mu_Y z} { \frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}} \right\} \\
\end{eqnarray} \)
第二，三項は，
\( \begin{eqnarray}
&&
- \frac{\frac{\sigma_X^2}{n_X}+\frac{\sigma_Y^2}{n_Y}}{2\frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}}
\left\{ - \left[ \frac{(\frac{\sigma_Y^2}{n_Y} \mu_X + \frac{\sigma_X^2}{n_X} z \color{red}{+} \frac{\sigma_X^2}{n_X} \mu_Y)}{\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}} \right]^2+ \frac{
\frac{\sigma_Y^2}{n_Y} \mu_X^2 + \frac{\sigma_X^2}{n_X} z^2 + \frac{\sigma_X^2}{n_X} \mu_Y^2 \color{red}{+} 2 \frac{\sigma_X^2}{n_X} \mu_Y z} { \frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}} \right\} \\
&=&
- \frac{\frac{\sigma_X^2}{n_X}+\frac{\sigma_Y^2}{n_Y}}{2\frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}}
\left\{ - \frac{(\frac{\sigma_Y^2}{n_Y} \mu_X + \frac{\sigma_X^2}{n_X} z \color{red}{+} \frac{\sigma_X^2}{n_X} \mu_Y)^2}{(\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y})^2} + \frac{
(\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y})(\frac{\sigma_Y^2}{n_Y} \mu_X^2 + \frac{\sigma_X^2}{n_X} z^2 + \frac{\sigma_X^2}{n_X} \mu_Y^2 \color{red}{+} 2 \frac{\sigma_X^2}{n_X} \mu_Y z)} { (\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y})^2} \right\} \\
&=& - \frac{\frac{\sigma_X^2}{n_X}+\frac{\sigma_Y^2}{n_Y}}{2\frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}} \left\{ \begin
{array}{1}
- \frac{\color{cyan}{\sigma_Y^4 \mu_X^2} + \color{red}{\sigma_X^4 z^2} +\color{green}{\sigma_X^4 \mu_Y^2} + 2 \frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y} \mu_X z \color{red}{+}2\frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y} \mu_X \mu_Y \color{blue}{\color{red}{+}2\sigma_X^4 \mu_Y z}}{(\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y})^2}\\
\frac{
\frac{\sigma_X^2}{n_X}\frac{\sigma_Y^2}{n_Y} \mu_X^2 + \color{red}{\sigma_X^4 z^2} + \color{green}{\sigma_X^4 \mu_Y^2} \color{red}{+} \color{blue}{2 \sigma_X^4 \mu_Y z}+\color{cyan}{\sigma_Y^4 \mu_X^2} + \frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}z^2 + \frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}\mu_Y^2 \color{red}{+} 2 \frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}\mu_Y z} { (\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y})^2}
\end{array} \right\}
\\
\end{eqnarray} \)
色の部分がキャンセルされるので，
\( \begin{eqnarray}
&=& - \frac{\frac{\sigma_X^2}{n_X}+\frac{\sigma_Y^2}{n_Y}}{2\frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}} \left\{
\frac{ -2 \frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y} \mu_X z \color{red}{-} 2\frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y} \mu_X \mu_Y +
\frac{\sigma_X^2}{n_X}\frac{\sigma_Y^2}{n_Y} \mu_X^2 + \frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}z^2 + \frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}\mu_Y^2 \color{red}{+} 2 \frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}\mu_Y z}{(\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y})^2}
\right\}
\\
&=& - \frac{\frac{\sigma_X^2}{n_X}+\frac{\sigma_Y^2}{n_Y}}{2\frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}} \frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y} \left\{
\frac{ -2 \mu_X z + 2 \mu_X \mu_Y +
\mu_X^2 + z^2 + \mu_Y^2 \color{red}{+} 2 \mu_Y z}{(\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y})^2}
\right\}
\\
&=& - \frac{ \{ z-(\mu_X \color{red}{-} \mu_Y) \}^2}{2(\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y})} \\
\end{eqnarray} \)
と簡単になる．
指数部分に戻すと，
\( \begin{eqnarray}
&& -\frac{\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}}{2\frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}} \left[ \color{red}{t}-\frac{(\frac{\sigma_Y^2}{n_Y} \mu_X + \frac{\sigma_X^2}{n_X} z \color{red}{+} \frac{\sigma_X^2}{n_X} \mu_Y)}{\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}} \right]^2
- \frac{ \{ z-(\mu_X \color{red}{-} \mu_Y) \}^2}{2(\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y})} \\
\end{eqnarray} \)
したがって，
\( \begin{eqnarray} P_{X-Y} (z)
&=& \frac{1}{2 \pi \sqrt{\frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}}} \displaystyle \int_{-\infty}^{ \infty }
exp \left[-\frac{\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}}{2\frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}} \left\{ \color{red}{t}-\frac{(\frac{\sigma_Y^2}{n_Y} \mu_X + \frac{\sigma_X^2}{n_X} z - \frac{\sigma_X^2}{n_X} \mu_Y)}{\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}} \right\}^2
- \frac{ \{ z-(\mu_X \color{red}{-} \mu_Y) \}^2}{2(\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y})} \right]dt
\\
&=& \frac{1}{2 \pi \sqrt{\frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}}} exp \left[- \frac{ \{ z-(\mu_X \color{red}{-} \mu_Y) \}^2}{2(\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y})} \right] \displaystyle \int_{-\infty}^{ \infty }
exp \left[-\frac{\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}}{2\frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}} \left\{ \color{red}{t}-\frac{(\frac{\sigma_Y^2}{n_Y} \mu_X + \frac{\sigma_X^2}{n_X} z \color{red}{+} \frac{\sigma_X^2}{n_X} \mu_Y)}{\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}} \right\}^2
\right]dt
\\
\end{eqnarray} \)
積分項目は，
\( \begin{eqnarray} && \displaystyle \int_{-\infty}^{ \infty }
exp \left[-\frac{\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}}{2\frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}} \left\{ \color{red}{t}-\frac{(\frac{\sigma_Y^2}{n_Y} \mu_X + \frac{\sigma_X^2}{n_X} z \color{red}{+} \frac{\sigma_X^2}{n_X} \mu_Y)}{\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}} \right\}^2
\right]dt
\\
&=& \displaystyle \int_{-\infty}^{ \infty }
exp \left[-a\{ t-b\}^2
\right]dt
\\
\end{eqnarray} \)
と簡単に記載することができ，
\( \Large t-b = w \)とすると，
\( \Large dt = dw \)
\( \begin{eqnarray}
&=& \displaystyle \int_{-\infty}^{ \infty }
exp \left[-a w^2
\right]dt
\\
&=& \sqrt{\frac{\pi}{a}} \\
\end{eqnarray} \)
したがって，
\( \begin{eqnarray} P_{X-Y} (z)
&=& \frac{1}{2 \pi \sqrt{\frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}}} exp \left[- \frac{ \{ z-(\mu_X \color{red}{-} \mu_Y) \}^2}{2(\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y})} \right]
\sqrt{ \frac{\pi (2 \frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y})}{
\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}}} \\
&=& \frac{1}{2 \pi \sqrt{\frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y}}} \sqrt{ \frac{\pi (2 \frac{\sigma_X^2}{n_X} \frac{\sigma_Y^2}{n_Y})}{
\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y}}} exp \left[- \frac{ \{ z-(\mu_X \color{red}{-} \mu_Y) \}^2}{2(\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y})} \right]
\\
&=& \frac{1}{ \sqrt{2 \pi(\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y})}} exp \left[- \frac{ \{ z-(\mu_X \color{red}{-} \mu_Y) \}^2}{2(\frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y})} \right]
\\
\end{eqnarray} \)
これは，
平均：\( \Large \mu_X \color{red}{-} \mu_Y \)
分散：\( \Large \frac{\sigma_X^2}{n_X} + \frac{\sigma_Y^2}{n_Y} \)
の正規分布である．