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'''This [[Collective Table of Formulas|Collective table of formulas]] is proudly sponsored'''<br> '''by the [http://www.facebook.com/hkn.beta Nice Guys of Eta Kappa Nu].''' <br><br> Visit us at the HKN Lounge in EE24 for hot coffee and fresh bagels only $1 each!
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'''This [[Collective Table of Formulas|Collective table of formulas]] is proudly sponsored'''<br> '''by the [http://www.facebook.com/hkn.beta Nice Guys of Eta Kappa Nu].''' <br><br> Visit us at the [[HKN|HKN Lounge]] in EE24 for hot coffee and fresh bagels only $1 each!
  
 
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Revision as of 07:00, 24 April 2012

This Collective table of formulas is proudly sponsored
by the Nice Guys of Eta Kappa Nu.

Visit us at the HKN Lounge in EE24 for hot coffee and fresh bagels only $1 each!

                                         HKNlogo.jpg


Probability Distributions
Random variable Probability density function $ f_{x}(x) $ Mean Variance Characteristic function $ \Phi_{x}(\omega) $
Normal or Gaussian $ N(\mu,\sigma^{2}) $ $ \dfrac{1}{\sqrt{2\pi\sigma^{2}}}e^{-(x-\mu)^{2}/2\sigma^{2}} $, $ -\infty<x<\infty $ $ \mu\ $ $ \sigma^{2}\ $ $ e^{j\mu\omega-\sigma^{2}\omega^{2}/2} $
Exponential $ E(\lambda) $ $ \lambda e^{-\lambda x},x\geq0,\lambda>0 $ $ \dfrac{1}{\lambda} $ $ \dfrac{1}{\lambda^{2}} $
Gamma $ G(\alpha,\beta) $ $ \dfrac{x^{\alpha-1}}{\Gamma(\alpha)\beta^{\alpha}}e^{-x/\beta},x\geq0,\alpha<0,\beta>0 $ $ \alpha\beta\ $ $ \alpha\beta^{2}\ $
Erlang- $ k $ $ \dfrac{(k\lambda)^{\lambda}}{(k-1)!}x^{k-1}e^{-k\lambda x} $ $ \dfrac{1}{\lambda} $ $ \dfrac{1}{k\lambda^{2}} $
Chi-square $ \chi^{2}(n) $ $ \dfrac{x^{n/2-1}}{2^{n/2}\Gamma(n/2)}e^{-x/2},x\geq0 $ $ n\ $ $ 2n\ $
Rayleigh $ \dfrac{x}{\sigma^{2}}e^{-x^{2}/2\sigma^{2}},x\geq0 $ $ \sqrt{\dfrac{\pi}{2}\sigma} $ $ (2-\pi/2)\sigma^{2}\ $
Uniform $ U(a,b) $ $ \dfrac{1}{b-a},a<x<b $ $ \dfrac{a+b}{2} $ $ \dfrac{(b-a)^{2}}{12} $
Beta $ \beta(\alpha,\beta) $ $ \dfrac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1},0<x<1,\alpha>0,\beta>0 $ $ \dfrac{\alpha}{\alpha+\beta} $ $ \dfrac{\alpha\beta}{(\alpha+\beta)^{2}(\alpha+\beta+1)} $
Cauchy $ \dfrac{\alpha/\pi}{(x-\mu)^{2}+\alpha^{2}} $ - $ \infty $ $ e^{j\omega\mu}e^{-\alpha|\omega|} $
Nakagami $ \dfrac{2}{\Gamma(m)}(\dfrac{m}{\Omega})^{m}x^{2m-1}e^{-\dfrac{m}{\Omega}x^{2}} $ $ \dfrac{\Gamma(m+1/2)}{\Gamma(m)}\sqrt{\dfrac{\Omega}{m}} $ $ \Omega(1-\dfrac{1}{m}(\dfrac{\Gamma(m+1/2)}{\Gamma(m)})^{2}) $
Students $ f(n) $ $ \dfrac{\Gamma((n+1)/2)}{\sqrt{\pi n}\Gamma(n/2)}(\dfrac{m}{n})^{m/2}x^{m/2-1}(1+\dfrac{mx}{n})^{-(m+n)/2},x>0 $ 0 $ \dfrac{n}{n-2},n>2 $
$ F- $ distribution $ \dfrac{\Gamma((n+1)/2)}{\sqrt{\pi n}\Gamma(n/2)}(\dfrac{m}{n})^{m/2}x^{m/2-1}(1+\dfrac{mx}{n})^{-(m+n)/2},x>0 $ $ \dfrac{n}{n-2},n>2 $ $ \dfrac{n^{2}(2m+2n-4)}{m(n-2)^{2}(n-4)},n>4 $
Bernoulli $ P(X=1)=p,P(X=0)=1-p=q\ $ $ p\ $ $ p(1-p)\ $ $ pe^{j\omega}+q\ $ \tabularnewline
Binomial $ B(n,p) $ $ (\binom{n}{k}p^{k}q^{n-k}), $ $ k=0,1,2,\cdots n,p+q=1 $ $ np\ $ $ npq\ $ $ (pe^{j\omega}+q)^{n} $ \tabularnewline
Poisson $ P(\lambda) $ $ \dfrac{e^{-\lambda}\lambda^{k}}{k!},k=0,1,2,\cdots,\infty $ $ \lambda\ $ $ \lambda\ $ $ e^{-\lambda(1-e^{j\omega})} $ \tabularnewline
Hypergeometric $ \dfrac{\binom{M}{k}\binom{N-M}{n-k}}{\binom{N}{n}}, $ $ max(0,M+n-N)\leq k\leq min(M,n) $ $ \dfrac{nM}{N} $ $ n\dfrac{M}{N}(1-\dfrac{M}{N})(1-\dfrac{n-1}{N-1}) $
Geometric $ \begin{cases} \dfrac{pq^{k},k=0,1,2\ldots,\infty}{pq^{k-1},k=1,2\ldots,\infty,p+q=1} | | .\end{cases} $ $ {\dfrac{q}{p}\atop \dfrac{1}{p}} $ $ \dfrac{q}{p^{2}} $ $ \dfrac{p}{1-qe^{j\omega}} $ or $ \dfrac{p}{e^{-j\omega}-q} $
Pascal or negative binomial $ NB(r,p) $ $ \begin{cases} \dfrac{\binom{r+k-1}{k}p^{r}q^{k},k=0,1,2,\ldots,\infty}{\binom{k-1}{r-1}p^{r}q^{k-r},k=r,r+1,\ldots,\infty,p+q=1} | | .\end{cases} $ $ {\dfrac{rq}{p}\atop \dfrac{r}{p}} $ $ \dfrac{rq}{p^{2}} $ $ (\dfrac{p}{1-qe^{-j\omega}})^{r} $ or( $ \dfrac{p}{e^{-j\omega}-q}) $
Discrete uniform $ 1/N,k=1,2,\ldots,N $ $ \dfrac{N+1}{2} $ $ \dfrac{N^{2}-1}{12} $ $ e^{j(N+1)\omega/2}\dfrac{sin(Nw/2)}{sin(\omega/2)} $

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