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)} $ | |||||
