Transformations for Discrete R.V. (MLPP 2.6.2.)

$$ p_y(y)=\sum_{x:f(x)=y}p_x(x) $$

Transformations for Continuous R.V. (MLPP 2.6.2.)

$$ p_y(y)=p_x(x)|\frac{dx}{dy}| $$

• Jacobian Matrix

  $$ J=\begin{bmatrix}\frac{\partial f_1}{\partial x_1} & \cdots & \frac{\partial f_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1} & \cdots & \frac{\partial f_m}{\partial x_n} \end{bmatrix} $$

• Taylor expansion: 1st approximation

  $$ f(x)=f(p)+J_f(x-p)+o(||x-p||) $$

Central Limit Theorem

$$ Z=\lim_{n \rightarrow \infin} \frac{\bar X_n-\mu}{\sigma_{\bar X}}, \sim N(0,1) \text{ with } \sigma_{\bar X}=\sigma/\sqrt n $$