User:TimothyChenAllen/PDFs

Distribution	PDF/PMF	Mean	Variance
Bernoulli $\,P(X=1)=p$	${\begin{cases}q=(1-p)&{\text{for }}k=0\\p&{\text{for }}k=1\end{cases}}$	$p\,$	$p(1-p)\,$
Geometric $(1-p)^{k-1}\,p\!$	$(1-p)^{k-1}\,p\!$	${\frac {1}{p}}\!$	${\frac {1-p}{p^{2}}}\!$
Binomial B(n, p)	$\textstyle {n \choose k}\,p^{k}(1-p)^{n-k}$	np	np(1 - p)
Poisson Pois(?)	${\frac {\lambda ^{k}}{k!}}\cdot e^{-\lambda }$	$\lambda \,\!$	$\lambda \,\!$
Uniform (continuous) U(a, b)	${\begin{cases}{\frac {1}{b-a}}&{\text{for }}x\in [a,b]\\0&{\text{otherwise}}\end{cases}}$	${\tfrac {1}{2}}(a+b)$	${\tfrac {1}{2}}(a+b)$
Uniform (discrete) U(a, b)	${\begin{matrix}{\frac {1}{n}}&{\mbox{for }}a\leq k\leq b\ \\0&{\mbox{otherwise }}\end{matrix}}$	${\frac {a+b}{2}}\,$	${\frac {n^{2}-1}{12}}$
Normal N(µ, s2)	${\frac {1}{\sigma {\sqrt {2\pi }}}}\,e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}$	µ	$\sigma ^{2}\,$
Chi-squared ?2k	${\frac {1}{2^{\frac {k}{2}}\Gamma \left({\frac {k}{2}}\right)}}\;x^{{\frac {k}{2}}-1}e^{-{\frac {x}{2}}}\,$	k	2k
Gamma G(k, ?)	$\scriptstyle {\frac {1}{\Gamma (k)\theta ^{k}}}x^{k\,-\,1}e^{-{\frac {x}{\theta }}}\,\!$	$\scriptstyle \operatorname {E} [X]=k\theta \!$ $\scriptstyle \operatorname {E} [\ln X]=\psi (k)+\ln(\theta )\!$ (see digamma function)	$\scriptstyle \operatorname {Var} [X]=k\theta ^{2}\,\!$ $\scriptstyle \operatorname {Var} [\ln X]=\psi _{1}(k)\!$ (see trigamma function )
Exponential Exp(?)	? e-?x	?-1	?-2
Multivariate normal N(µ, S)	$(2\pi )^{-{\frac {k}{2}}}|{\boldsymbol {\Sigma }}|^{-{\frac {1}{2}}}\,e^{-{\frac {1}{2}}(\mathbf {x} -{\boldsymbol {\mu }})'{\boldsymbol {\Sigma }}^{-1}(\mathbf {x} -{\boldsymbol {\mu }})},$ exists only when S is positive-definite	µ	S
Degenerate da	${\begin{matrix}1&{\mbox{for }}k=k_{0}\\0&{\mbox{otherwise }}\end{matrix}}$	$k_{0}\,$	$0\,$
Laplace L(µ, b)	${\frac {1}{2\,b}}\exp \left(-{\frac {|x-\mu |}{b}}\right)\,$	µ	2b2
Negative Binomial NB(r, p)	${k+r-1 \choose k}\cdot (1-p)^{r}p^{k},\!$involving a binomial coefficient	${\frac {pr}{1-p}}$	${\frac {pr}{(1-p)^{2}}}$
Cauchy Cauchy(µ, ?)	${\frac {1}{\pi \gamma \,\left[1+\left({\frac {x-x_{0}}{\gamma }}\right)^{2}\right]}}\!$	undefined	undefined