第2講 確率分布

$$\begin{equation} f_{X,Y}(x,y) = \frac{1}{2\pi\sigma_X\sigma_Y\sqrt{1-\rho^2}} \exp\left[-\frac{1}{2(1-\rho^2)}\left\{\frac{(x - \mu_X)^2}{{\sigma_X}^2} - \frac{2\rho(x - \mu_X)(y - \mu_Y)}{\sigma_X\sigma_Y} + \frac{(y - \mu_Y)^2}{{\sigma_Y}^2}\right\}\right] \end{equation}$$

$$\begin{equation} \rho = \frac{\sigma_{XY}}{\sigma_X\,\sigma_Y} \end{equation}$$

$$\begin{align} f_{X}(x) &= \int_{-\infty}^{\infty}f_{X,Y}(x,y)\,dy \\ &= \frac{1}{\sqrt{2\pi}\sigma_X}\exp\left[- \frac{(x - \mu_X)^2}{{2\,\sigma_X}^2}\right] \int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi}\sqrt{(1 - \rho^2){\sigma_{Y}}^2}}\exp\left[-\frac{1}{2(1 - \rho^2){\sigma_{Y}}^2}\left\{y - \mu_{Y} - \rho\frac{\sigma_Y}{\sigma_{X}}(x - \mu_{X})\right\}^2\right]\,dy \\ &= \frac{1}{\sqrt{2\pi}\sigma_X}\exp\left[- \frac{(x - \mu_X)^2}{{2\,\sigma_X}^2}\right] \end{align}$$

$$\begin{equation} X \sim N(\mu_X, {\sigma_{X}}^2) \end{equation}$$

$$\begin{equation} Y \sim N(\mu_Y, {\sigma_{Y}}^2) \end{equation}$$

$$\begin{equation} x = \mu_X + \sigma_X u \end{equation}$$

$$\begin{equation} y = \mu_Y + \sigma_Y v \end{equation}$$

$$\begin{equation} \begin{vmatrix} \dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v} \\ \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} \end{vmatrix} = \begin{vmatrix} \sigma_X & 0 \\ 0 & \sigma_Y \end{vmatrix} = \sigma_X \sigma_Y \end{equation}$$

$$\begin{align} Cov[X,Y] &= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}(x - \mu_X)(y - \mu_Y)\frac{1}{2\pi\sigma_X\sigma_Y\sqrt{1-\rho^2}}\exp\left[-\frac{1}{2(1-\rho^2)}\left\{\frac{(x - \mu_X)^2}{{\sigma_X}^2} - \frac{2\rho(x - \mu_X)(y - \mu_Y)}{\sigma_X\sigma_Y} + \frac{(y - \mu_Y)^2}{{\sigma_Y}^2}\right\}\right]\,dxdy \\ &= \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\sigma_X u \sigma_Y v\frac{1}{2\pi\sigma_X\sigma_Y\sqrt{1-\rho^2}}\exp\left[-\frac{1}{2(1-\rho^2)}\left\{u^2 - 2\rho u v + v^2\right\}\right]\cdot \sigma_X \sigma_Y\,dudv \\ &= \sigma_X \sigma_Y\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}u v\frac{1}{2\pi\sqrt{1-\rho^2}}\exp\left[-\frac{1}{2(1-\rho^2)}\left\{u^2 - (\rho u)^2 + (\rho u)^2 - 2\rho u v + v^2\right\}\right]\,dudv \\ &= \sigma_X \sigma_Y\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}u v\frac{1}{2\pi\sqrt{1-\rho^2}}\exp\left[-\frac{1}{2(1-\rho^2)}\left\{(1 - \rho^2)u^2 + (v - \rho u)^2\right\}\right]\,dudv \\ &= \sigma_X \sigma_Y\int_{-\infty}^{\infty}u \frac{1}{\sqrt{2\pi}}\exp\left[-\frac{u^2}{2}\right]\left\{\int_{-\infty}^{\infty}v\frac{1}{\sqrt{2\pi}\sqrt{1-\rho^2}}\exp\left[-\frac{(v - \rho u)^2}{2(1-\rho^2)}\right]\,dv\right\}\,dv \\ &= \sigma_X \sigma_Y\int_{-\infty}^{\infty}u \frac{1}{\sqrt{2\pi}}\exp\left[-\frac{u^2}{2}\right] \cdot \rho u\,dv \\ &= \sigma_X \sigma_Y \rho \int_{-\infty}^{\infty}u^2 \frac{1}{\sqrt{2\pi}}\exp\left[-\frac{u^2}{2}\right]\,dv \\ &= \sigma_X \sigma_Y \rho \end{align}$$

$$\begin{equation} r_{XY} = \frac{Cov[X, Y]}{V[X]V[Y]} = \frac{\sigma_X \sigma_Y \rho}{\sigma_X \sigma_Y} = \rho \end{equation}$$

$$\begin{align} f_{Y \mid X}(x) &= \frac{f_{X,Y}(x,y)}{f_{X}(x)} \\ &= \frac{1}{\sqrt{2\pi}(\sigma_Y\sqrt{1-\rho^2})}\exp\left[-\frac{1}{2(\sigma_Y\sqrt{1-\rho^2})^2}\left\{y - \left(\mu_Y + \frac{\sigma_Y}{\sigma_X}\rho(x - \mu_X)\right)\right\}^2\right] \end{align}$$

$$\begin{equation} Y \mid X~~\sim~~N\left(\mu_Y + \frac{\sigma_Y}{\sigma_X}\rho(X - \mu_X),~{\sigma_Y}^2(1-\rho^2)\right) \\ \end{equation}$$

$$\begin{equation} Y \mid X~\sim~ N\left( \mu_Y + \frac{\sigma_{XY}}{{\sigma_X}^2}(X - \mu_X),~ {\sigma_Y}^2 - \frac{{\sigma_{XY}}^2}{{\sigma_X}^2} \right) \end{equation}$$