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

• Calculus
  • $\text{exp}(y)=e^y=1+y+\frac{y^2}{2!}+\cdots +\frac{y^{\mathrm{\infty }}}{\mathrm{\infty }!}$
  • $\underset{a}{\overset{b}{\int }}u\delta v=uv{|}_a^b-\underset{a}{\overset{b}{\int }}v\delta u$
  • $\mathrm{\Gamma }(\alpha )=\underset{0}{\overset{\mathrm{\infty }}{\int }}{y}^{\alpha -1}{e}^{-y}\delta y$
  • $\mathrm{\Gamma }(1/2)=\sqrt{\pi },\mathrm{\Gamma }(1)=1,\mathrm{\Gamma }(\alpha +1)=\alpha \mathrm{\Gamma }(\alpha )$
• Moment Formulas
  • $M_X(u)=E\left[e^{uX}\right]$
  • $\text{Var}(X)=E\left[X^2\right]-{\left(E\left[X\right]\right)}^2$
• Central Limit Theorem
  • $P(a<\frac{\sqrt{m}(\hat{\mathrm{\Theta }}\left(m\right)-\mathrm{\Theta })}{\rho }\le b)\begin{array}{c}{}_{m\to \mathrm{\infty }}\\ \overrightarrow{}\end{array}\underset{a}{\overset{b}{\int }}\frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi }}\delta x$
• Convolution
  • $f_{X+Y}(y)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{f}_X(y-z)f_Y(z)\delta z=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{f}_X(z)f_Y(y-z)\delta z$
• Reliability and Hazard
  • $R_S(t)=\prod _{i=1}^N{R}_i(t)$
  • $R_P(t)=1-\prod _{i=1}^N(1-R_i(t))$
  • $h_T(t)=\frac{f_T(t)}{R_T(t)}$
  • $R_T(t)=\text{exp}(-\underset{0}{\overset{t}{\int }}{h}_T(s)\delta s)$
• Redundancy
  • $R(t)=\sum _{i=k}^m(\genfrac{}{}{0ex}{}{m}{i}){(R_a(t))}^i{(1-R_a(t))}^{m-i}$