第1講 確率と確率変数

$$\begin{equation} \varSigma = Var[\bm{x}] = E\left[(\bm{x} - \bm{\mu})(\bm{x} - \bm{\mu})^T\right] \end{equation}$$

$$\begin{align} \varSigma &= E\left[(\bm{x} - \bm{\mu})(\bm{x} - \bm{\mu})^T\right] \\ &= E\left[\bm{x}\bm{x}^T - \bm{x}\bm{\mu}^T - \bm{\mu}\bm{x}^T + \bm{\mu}\bm{\mu}^T\right] \\ &= E\left[\bm{x}\bm{x}^T\right] - E\left[\bm{x}\right]\bm{\mu}^T - \bm{\mu}E\left[\bm{x}^T\right] + \bm{\mu}\bm{\mu}^T \\ &= E\left[\bm{x}\bm{x}^T\right] - \bm{\mu}\bm{\mu}^T - \bm{\mu}\bm{\mu}^T + \bm{\mu}\bm{\mu}^T \\ &= E\left[\bm{x}\bm{x}^T\right] - \bm{\mu}\bm{\mu}^T \end{align}$$

$$\begin{align} & E\left[\bm{x}^TA\bm{x}\right] \\ &\quad = E\left[\mathrm{tr}(\bm{x}^TA\bm{x})\right] \\ &\quad = E\left[\mathrm{tr}(A\bm{x}\bm{x}^T)\right] \\ &\quad = \mathrm{tr}\left(E\left[A\bm{x}\bm{x}^T\right]\right) \\ &\quad = \mathrm{tr}\left(A E\left[\bm{x}\bm{x}^T\right]\right) \\ &\quad = \mathrm{tr}\left(A (\varSigma + \bm{\mu}\bm{\mu}^T)\right) \\ &\quad = \mathrm{tr}\left(A \varSigma + A \bm{\mu}\bm{\mu}^T\right) \\ &\quad = \mathrm{tr}\left(A \varSigma\right) + \mathrm{tr}\left(A \bm{\mu}\bm{\mu}^T\right) \\ &\quad = \mathrm{tr}\left(A \varSigma\right) + \mathrm{tr}\left(\bm{\mu}^T A \bm{\mu}\right) \\ &\quad = \mathrm{tr}\left(A \varSigma\right) + \bm{\mu}^T A \bm{\mu} \end{align}$$