Probability Distributions

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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