Collective Table of Formulas

Probability Distributions

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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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