第1講確率と確率変数

平均ベクトル

確率変数がベクトルの場合

μ = E [x] = (\begin{matrix} E [x_{1}] \\ ⋮ \\ E [x_{n}] \end{matrix})

$\begin{equation} \bm{\mu} = E[\bm{x}] = \begin{pmatrix} E[x_1] \\ \vdots \\ E[x_n] \end{pmatrix} \end{equation}$

E [a^{T} x] = a^{T} E [x]

$\begin{equation} E[\bm{a}^T\bm{x}] = \bm{a}^T\,E[\bm{x}] \end{equation}$

E [x^{T} a] = E [a^{T} x]

$\begin{equation} E[\bm{x}^T\bm{a}] = E[\bm{a}^T\bm{x}] \end{equation}$

E [A x + b] = A μ + b

$\begin{equation} E[A\bm{x} + \bm{b}] = A\bm{\mu} + \bm{b} \end{equation}$

E [x^{T} A + b^{T}] = μ^{T} A + b^{T}

$\begin{equation} E[\bm{x}^TA + \bm{b}^T] = \bm{\mu}^TA + \bm{b}^T \end{equation}$

確率変数が行列の場合

μ = E [X] = (\begin{matrix} E [x_{11}] & \dots & E [x_{1 n}] \\ ⋮ & ⋱ & ⋮ \\ E [x_{n 1}] & \dots & E [x_{n n}] \end{matrix})

$\begin{equation} \bm{\mu} = E[X] = \begin{pmatrix} E[x_{11}] & \cdots & E[x_{1n}] \\ \vdots & \ddots & \vdots \\ E[x_{n1}] & \cdots & E[x_{nn}] \end{pmatrix} \end{equation}$

E [A X] = A E [X]

$\begin{equation} E[AX] = A\,E[X] \end{equation}$

E [X B] = E [X] B

$\begin{equation} E[XB] = E[X]\,B \end{equation}$

E [A X B] = A E [X] B

$\begin{equation} E[AXB] = A\,E[X]\,B \end{equation}$

第1講 確率と確率変数

平均ベクトル

確率変数がベクトルの場合

確率変数が行列の場合

第1講確率と確率変数