0.6 Mathematical phrases, symbols, and formulas

English phrases written mathematically

Formulas

Formula 1: factorial

$0!=1$

Formula 2: combinations

Formula 3: binomial distribution

$P(X=x)=\left(\begin{array}{c}n\\ x\end{array}\right)p^x{q}^{n-x}$ , for $x=0,1,2,...,n$

Formula 4: geometric distribution

$P(X=x)=q^{x-1}p$ , for $x=1,2,3,...$

Formula 5: hypergeometric distribution

$P\text{(}X=x\text{)}=\left(\frac{\left(\genfrac{}{}{0ex}{}{r}{x}\right)\left(\genfrac{}{}{0ex}{}{b}{n-x}\right)}{\left(\genfrac{}{}{0ex}{}{r+b}{n}\right)}\right)$

Formula 6: poisson distribution

$P\text{(}X=x\text{)}=\frac{{\mu }^x{e}^{-\mu }}{x!}$

Formula 7: uniform distribution

$f(X)=\frac{1}{b-a}$ , $a<x<b$

Formula 8: exponential distribution

$f(x)=m{e}^{-mx}m>0,x\ge 0$

Formula 9: normal distribution

$f\text{(}x\text{)}=\frac{1}{\sigma \sqrt{2\pi }}{e}^{\frac{{-(x-\mu )}^2}{{2\sigma }^2}}$ , $–\infty <x<\infty$

Formula 10: gamma function

$\Gamma \left(\frac{1}{2}\right)=\sqrt{\pi }$

$\Gamma (m+1)=m!$ for $m$ , a nonnegative integer

otherwise: $\Gamma (a+1)=a\Gamma (a)$

Formula 11: student's t -distribution

$f\text{(}x\text{)}=\frac{{\left(1+\frac{x^2}{n}\right)}^{\frac{-(n+1)}{2}}\Gamma \left(\frac{n+1}{2}\right)}{\sqrt{\mathrm{n\pi }}\Gamma \left(\frac{n}{2}\right)}$

$X=\frac{Z}{\sqrt{\frac{Y}{n}}}$

$Z~N(0,1),Y~{{\rm X}}_{df}^2$ , $n$ = degrees of freedom

Formula 12: chi-square distribution

$f\text{(}x\text{)}=\frac{x^{\frac{n-2}{2}}{e}^{\frac{-x}{2}}}{2^{\frac{n}{2}}\Gamma \left(\frac{n}{2}\right)}$ , $x>0$ , $n$ = positive integer and degrees of freedom

Formula 13: f distribution

$df(n)=$ degrees of freedom for the numerator

$df(d)=$ degrees of freedom for the denominator

$f(x)=\frac{\Gamma (\frac{u+v}{2})}{\Gamma (\frac{u}{2})\Gamma (\frac{v}{2})}{(\frac{u}{v})}^{\frac{u}{2}}{x}^{(\frac{u}{2}-1)}[1+(\frac{u}{v})x^{-0.5(u+v)}]$

$X=\frac{Y_u}{W_v}$ , $Y$ , $W$ are chi-square

Symbols and their meanings