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
  • $R(t)=\sum _{i=k}^{m}{m \choose i}\left(R_{a}(t)\right)^{i}\left(1-R_{a}(t)\right)^{m-i}$