User:TimothyChenAllen/Probability Distributions

Distribution	Moment-generating function MX(t)	Characteristic function φ(t)	PDF/PMF	CDF	Mean	Variance
Bernoulli $\,P(X=1)=p$	$\,1-p+pe^{t}$	$\,1-p+pe^{it}$	${\begin{cases}q=(1-p)&{\text{for }}k=0\\p&{\text{for }}k=1\end{cases}}$	${\begin{cases}0&{\text{for }}k<0\\q&{\text{for }}0\leq k<1\\1&{\text{for }}k\geq 1\end{cases}}$	$p\,$	$p(1-p)\,$
Geometric $(1-p)^{k-1}\,p\!$	${\frac {pe^{t}}{1-(1-p)e^{t}}}\!$, for  $t<-\ln(1-p)\!$	${\frac {pe^{it}}{1-(1-p)\,e^{it}}}\!$	$(1-p)^{k-1}\,p\!$	$1-(1-p)^{k}\!$	${\frac {1}{p}}\!$	${\frac {1-p}{p^{2}}}\!$
Binomial B(n, p)	$\,(1-p+pe^{t})^{n}$	$\,(1-p+pe^{it})^{n}$	$\textstyle {n \choose k}\,p^{k}(1-p)^{n-k}$	$F(x;n,p)=\Pr(X\leq x)=\sum _{i=0}^{\lfloor x\rfloor }{n \choose i}p^{i}(1-p)^{n-i}$	np	np(1 − p)
Poisson Pois(λ)	$\,e^{\lambda (e^{t}-1)}$	$\,e^{\lambda (e^{it}-1)}$	${\frac {\lambda ^{k}}{k!}}\cdot e^{-\lambda }$	${\frac {\Gamma (\lfloor k+1\rfloor ,\lambda )}{\lfloor k\rfloor !}}\!$ --or-- $e^{-\lambda }\sum _{i=0}^{\lfloor k\rfloor }{\frac {\lambda ^{i}}{i!}}\$ (for $k\geq 0$ where $\Gamma (x,y)\,\!$ is the Incomplete gamma function and $\lfloor k\rfloor$ is the floor function)	$\lambda \,\!$	$\lambda \,\!$
Uniform (continuous) U(a, b)	$\,{\frac {e^{tb}-e^{ta}}{t(b-a)}}$	$\,{\frac {e^{itb}-e^{ita}}{it(b-a)}}$	${\begin{cases}{\frac {1}{b-a}}&{\text{for }}x\in [a,b]\\0&{\text{otherwise}}\end{cases}}$	${\begin{cases}0&{\text{for }}x<a\\{\frac {x-a}{b-a}}&{\text{for }}x\in [a,b)\\1&{\text{for }}x\geq b\end{cases}}$	${\tfrac {1}{2}}(a+b)$	${\tfrac {1}{2}}(a+b)$
Uniform (discrete) U(a, b)	$\,{\frac {e^{at}-e^{(b+1)t}}{(b-a+1)(1-e^{t})}}$	$\,{\frac {e^{ait}-e^{(b+1)it}}{(b-a+1)(1-e^{it})}}$	${\begin{matrix}{\frac {1}{n}}&{\mbox{for }}a\leq k\leq b\ \\0&{\mbox{otherwise }}\end{matrix}}$	${\begin{matrix}0&{\mbox{for }}k<a\\{\frac {\lfloor k\rfloor -a+1}{n}}&{\mbox{for }}a\leq k\leq b\\1&{\mbox{for }}k>b\end{matrix}}$	${\frac {a+b}{2}}\,$	${\frac {n^{2}-1}{12}}$
Normal N(μ, σ2)	$\,e^{t\mu +{\frac {1}{2}}\sigma ^{2}t^{2}}$	$\,e^{it\mu -{\frac {1}{2}}\sigma ^{2}t^{2}}$	${\frac {1}{\sigma {\sqrt {2\pi }}}}\,e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}$	${\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sqrt {2\sigma ^{2}}}}\right)\right]$	μ	$\sigma ^{2}\,$
Chi-squared χ2k	$\,(1-2t)^{-k/2}$	$\,(1-2it)^{-k/2}$	${\frac {1}{2^{\frac {k}{2}}\Gamma \left({\frac {k}{2}}\right)}}\;x^{{\frac {k}{2}}-1}e^{-{\frac {x}{2}}}\,$	${\frac {1}{\Gamma \left({\frac {k}{2}}\right)}}\;\gamma \left({\frac {k}{2}},\,{\frac {x}{2}}\right)$	k	2k
Gamma Γ(k, θ)	$\,(1-t\theta )^{-k}$	$\,(1-it\theta )^{-k}$	$\scriptstyle {\frac {1}{\Gamma (k)\theta ^{k}}}x^{k\,-\,1}e^{-{\frac {x}{\theta }}}\,\!$	$\scriptstyle {\frac {1}{\Gamma (k)}}\gamma \left(k,\,{\frac {x}{\theta }}\right)\!$	$\scriptstyle \operatorname {E} [X]=k\theta \!$ $\scriptstyle \operatorname {E} [\ln X]=\psi (k)+\ln(\theta )\!$ (see digamma function)	$\scriptstyle \operatorname {Var} [X]=k\theta ^{2}\,\!$ $\scriptstyle \operatorname {Var} [\ln X]=\psi _{1}(k)\!$ (see trigamma function )
Exponential Exp(λ)	$\,(1-t\lambda ^{-1})^{-1}$	$\,(1-it\lambda ^{-1})^{-1}$	λ e−λx	1 − e−λx	λ−1	λ−2
Multivariate normal N(μ, Σ)	$\,e^{t^{\mathrm {T} }\mu +{\frac {1}{2}}t^{\mathrm {T} }\Sigma t}$	$\,e^{it^{\mathrm {T} }\mu -{\frac {1}{2}}t^{\mathrm {T} }\Sigma t}$	$(2\pi )^{-{\frac {k}{2}}}|{\boldsymbol {\Sigma }}|^{-{\frac {1}{2}}}\,e^{-{\frac {1}{2}}(\mathbf {x} -{\boldsymbol {\mu }})'{\boldsymbol {\Sigma }}^{-1}(\mathbf {x} -{\boldsymbol {\mu }})},$ exists only when Σ is positive-definite	(no analytic expression)	μ	Σ
Degenerate δa	$\,e^{ta}$	$\,e^{ita}$	${\begin{matrix}1&{\mbox{for }}k=k_{0}\\0&{\mbox{otherwise }}\end{matrix}}$	${\begin{matrix}0&{\mbox{for }}k<k_{0}\\1&{\mbox{for }}k\geq k_{0}\end{matrix}}$	$k_{0}\,$	$0\,$
Laplace L(μ, b)	$\,{\frac {e^{t\mu }}{1-b^{2}t^{2}}}$	$\,{\frac {e^{it\mu }}{1+b^{2}t^{2}}}$	${\frac {1}{2\,b}}\exp \left(-{\frac {|x-\mu |}{b}}\right)\,$	$\left\{{\begin{matrix}&{\frac {1}{2}}\exp \left({\frac {x-\mu }{b}}\right)&{\mbox{if }}x<\mu \\[8pt]1-\!\!\!\!&{\frac {1}{2}}\exp \left(-{\frac {x-\mu }{b}}\right)&{\mbox{if }}x\geq \mu \end{matrix}}\right.$	μ	2b2
Negative Binomial NB(r, p)	$\,{\frac {((1-p)e^{t})^{r}}{(1-pe^{t})^{r}}}$	$\,{\frac {((1-p)e^{it})^{r}}{(1-pe^{it})^{r}}}$	${k+r-1 \choose k}\cdot (1-p)^{r}p^{k},\!$involving a binomial coefficient	$1-I_{p}(k+1,\,r),$ the regularized incomplete beta function	${\frac {pr}{1-p}}$	${\frac {pr}{(1-p)^{2}}}$
Cauchy Cauchy(μ, θ)	does not exist	$\,e^{it\mu -\theta |t|}$	${\frac {1}{\pi \gamma \,\left[1+\left({\frac {x-x_{0}}{\gamma }}\right)^{2}\right]}}\!$	${\frac {1}{\pi }}\arctan \left({\frac {x-x_{0}}{\gamma }}\right)+{\frac {1}{2}}\!$	undefined	undefined