University of Alberta Guide/STAT/222/Formulas and Functions/General Formulas


 * Calculus
 * $$\mbox{exp}(y) = e^{y} = 1 + y + \frac{y^2}{2!} + \cdots + \frac{y^\infty}{\infty!}\,$$
 * $$\int_{a}^{b}u\delta v = uv|_{a}^{b} - \int_{a}^{b}v\delta u$$
 * $$\Gamma(\alpha) = \int_{0}^{\infty}y^{\alpha - 1}e^{-y}\delta y$$
 * $$\Gamma(1/2) = \sqrt{\pi}, \Gamma(1) = 1, \Gamma(\alpha + 1) = \alpha\Gamma(\alpha)$$
 * Moment Formulas
 * $$M_{X}(u) = E\left[e^{uX}\right]$$
 * $$\mbox{Var}(X) = E\left[X^{2}\right] - \left(E\left[X\right]\right)^{2}$$
 * Central Limit Theorem
 * $$P\left(a < \frac{\sqrt{m}\left(\hat{\Theta}\left(m\right) - \Theta\right)}{\rho} \leq b\right) \begin{matrix} {}_{m \rightarrow \infty} \\ \overrightarrow{\qquad\qquad} \\\end{matrix} \int_{a}^{b}\frac{e^{-\frac{x^{2}}{2}}}{\sqrt{2\pi}}\delta x$$
 * Convolution
 * $$f_{X + Y}(y) = \int_{-\infty}^{\infty}f_{X}(y - z)f_{Y}(z)\delta z = \int_{-\infty}^{\infty}f_{X}(z)f_{Y}(y - z)\delta z$$
 * Reliability and Hazard
 * $$R_{S}(t) = \prod_{i = 1}^{N}R_{i}(t)$$
 * $$R_{P}(t) = 1 - \prod_{i = 1}^{N}\left(1 - R_{i}(t)\right)$$
 * $$h_{T}(t) = \frac{f_{T}(t)}{R_{T}(t)}$$
 * $$R_{T}(t) = \mbox{exp}\left(- \int_{0}^{t}h_{T}(s)\delta s\right)$$
 * Redundancy
 * $$R(t) = \sum_{i = k}^{m}{m \choose i}\left(R_{a}(t)\right)^{i}\left(1 - R_{a}(t)\right)^{m - i}$$