Distribution

If $X \sim N(0,1)$, then $\mathbb{Z} = X^2 \sim \mathcal{X}^2(1)$

Beta($\alpha$, $\beta$) $0 < x < 1$

Uniform ($\alpha$, $\beta$) $a < x < b$

Gamma $\gamma(\alpha, \beta)$

$f(x) = {x^{\alpha - 1} \over \Gamma(\alpha) \theta^\alpha} e^{-x \over \theta}$

Exponential($\theta$) = $\gamma(\alpha = 1, theta)$

Chi-squared($\gamma$) = $\gamma(\alpha = {\gamma \over 2}, \theta = 2)$

$\mathcal{X} \sim N(\mu, \sigma_x^2)$ → $1 \over \sqrt{2 \pi} \sigma} e^{-{(x-\mu)^2} \over {2\sigma_x^2}$

$Y=g(x)$

$f_y(y) = \sum\limits_{x_k} f_x(x_k) |{dx \over dy}|_{\text{@} x = x_k}$