第1講 確率と確率変数

$$\begin{equation} V[Y] = E[V[Y \mid X]] + V[E[Y \mid X]] \end{equation}$$

$$\begin{align} V[Y] &= E[Y^2] - (E[Y])^2 \\ &= E[E[Y^2 \mid X]] - (E[E[Y \mid X]])^2 \\ &= E\left[V[Y \mid X] + (E[Y \mid X])^2\right] - (E[E[Y \mid X]])^2 \\ &= E[V[Y \mid X]] + E[(E[Y \mid X])^2] - (E[E[Y \mid X]])^2 \\ &= E[V[Y \mid X]] + V[E[Y \mid X]] \\ \end{align}$$

$$\begin{equation} X \sim N(\mu, \sigma^2) \end{equation}$$

$$\begin{equation} Y \mid X = x \sim N(ax + b, x^2) \end{equation}$$

$$\begin{align} E[Y] &= E[E[Y \mid X]] \\ &= E[aX + b] \\ &= aE[X] + b \\ &= a\mu + b \end{align}$$

$$\begin{align} V[Y] &= V[E[Y \mid X]] + E[V[Y \mid X]] \\ &= V[aX + b] + E[X^2] \\ &= a^2V[X] + V[X] + (E[X])^2 \\ &= a^2\sigma ^2 + \sigma ^2 + \mu^2 \\ &= (a^2 + 1)\sigma^2 + \mu^2 \\ \end{align}$$