TEMPLATE:INFOBOX PROBABILITY DISTRIBUTION TESTCASES

Binomial distribution
	Probability mass function
	Cumulative distribution function
Notation
Parameters	– number of trials; – success probability for each trial
Support	– number of successes
PMF
CDF
Mean
Median	or
Mode	or
Variance
Skewness
Excess kurtosis
Entropy	; in shannons. For nats, use the natural log in the log.
MGF
CF
PGF
Fisher information	; (for fixed )

Normal Distribution
	Probability density function ; The red curve is the standard normal distribution
	Cumulative distribution function
Notation
Parameters	= mean ( location); = variance (squared scale)
Support
PDF
CDF
Quantile
Mean
Median
Mode
Variance
MAD
Skewness
Ex. kurtosis
Entropy
MGF
CF
Fisher information	;
Kullback-Leibler divergence

Normal Distribution
	Probability density function ; The red curve is the standard normal distribution
	Cumulative distribution function
Notation
Parameters	= mean ( location); = variance (squared scale)
Support
PDF
CDF
Quantile
Mean
Median
Mode
Variance
MAD
Skewness
Excess kurtosis
Entropy
MGF
CF
Fisher information	;
Kullback–Leibler divergence

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Normal distribution

Side by side comparison

Side by side comparison

Binomial distribution
Probability mass function
Cumulative distribution function
Notation	$B(n,p)$
Parameters	$n\in \{0,1,2,\ldots \}$ – number of trials $p\in [0,1]$ – success probability for each trial
Support	$k\in \{0,1,\ldots ,n\}$ – number of successes
PMF	${\binom {n}{k}}p^{k}(1-p)^{n-k}$
CDF	$I_{1-p}(n-k,1+k)$
Mean	$np$
Median	$\lfloor np\rfloor$ or $\lceil np\rceil$
Mode	$\lfloor (n+1)p\rfloor$ or $\lceil (n+1)p\rceil -1$
Variance	$np(1-p)$
Skewness	${\frac {1-2p}{\sqrt {np(1-p)}}}$
Ex. kurtosis	${\frac {1-6p(1-p)}{np(1-p)}}$
Entropy	${\frac {1}{2}}\log _{2}\left(2\pi enp(1-p)\right)+O\left({\frac {1}{n}}\right)$ in shannons. For nats, use the natural log in the log.
MGF	$(1-p+pe^{t})^{n}$
CF	$(1-p+pe^{it})^{n}$
PGF	$G(z)=[(1-p)+pz]^{n}$
Fisher information	$g_{n}(p)={\frac {n}{p(1-p)}}$ (for fixed $n$ )

Side by side comparison

Geometric
Probability mass function
Cumulative distribution function
Parameters	$0<p<1$ success probability ( real)	$0<p\leq 1$ success probability ( real)
Support	k trials where $k\in \{1,2,3,\dots \}$	k failures where $k\in \{0,1,2,3,\dots \}$
PMF	$(1-p)^{k-1}p$	$(1-p)^{k}p$
CDF	$1-(1-p)^{k}$	$1-(1-p)^{k+1}$
Mean	${\frac {1}{p}}$	${\frac {1-p}{p}}$
Median	$\left\lceil {\frac {-1}{\log _{2}(1-p)}}\right\rceil$ (not unique if $-1/\log _{2}(1-p)$ is an integer)	$\left\lceil {\frac {-1}{\log _{2}(1-p)}}\right\rceil -1$ (not unique if $-1/\log _{2}(1-p)$ is an integer)
Mode	$1$	$0$
Variance	${\frac {1-p}{p^{2}}}$	${\frac {1-p}{p^{2}}}$
Skewness	${\frac {2-p}{\sqrt {1-p}}}$	${\frac {2-p}{\sqrt {1-p}}}$
Excess kurtosis	$6+{\frac {p^{2}}{1-p}}$	$6+{\frac {p^{2}}{1-p}}$
Entropy	${\tfrac {-(1-p)\log _{2}(1-p)-p\log _{2}p}{p}}$	${\tfrac {-(1-p)\log _{2}(1-p)-p\log _{2}p}{p}}$
MGF	${\frac {pe^{t}}{1-(1-p)e^{t}}},$ for $t<-\ln(1-p)$	${\frac {p}{1-(1-p)e^{t}}}$
CF	${\frac {pe^{it}}{1-(1-p)e^{it}}}$	${\frac {p}{1-(1-p)e^{it}}}$

Geometric
Probability mass function
Cumulative distribution function
Parameters	$0<p<1$ success probability ( real)	$0<p\leq 1$ success probability ( real)
Support	k trials where $k\in \{1,2,3,\dots \}$	k failures where $k\in \{0,1,2,3,\dots \}$
PMF	$(1-p)^{k-1}p$	$(1-p)^{k}p$
CDF	$1-(1-p)^{k}$	$1-(1-p)^{k+1}$
Mean	${\frac {1}{p}}$	${\frac {1-p}{p}}$
Median	$\left\lceil {\frac {-1}{\log _{2}(1-p)}}\right\rceil$ (not unique if $-1/\log _{2}(1-p)$ is an integer)	$\left\lceil {\frac {-1}{\log _{2}(1-p)}}\right\rceil -1$ (not unique if $-1/\log _{2}(1-p)$ is an integer)
Mode	$1$	$0$
Variance	${\frac {1-p}{p^{2}}}$	${\frac {1-p}{p^{2}}}$
Skewness	${\frac {2-p}{\sqrt {1-p}}}$	${\frac {2-p}{\sqrt {1-p}}}$
Ex. kurtosis	$6+{\frac {p^{2}}{1-p}}$	$6+{\frac {p^{2}}{1-p}}$
Entropy	${\tfrac {-(1-p)\log _{2}(1-p)-p\log _{2}p}{p}}$	${\tfrac {-(1-p)\log _{2}(1-p)-p\log _{2}p}{p}}$
MGF	${\frac {pe^{t}}{1-(1-p)e^{t}}},$ for $t<-\ln(1-p)$	${\frac {p}{1-(1-p)e^{t}}}$
CF	${\frac {pe^{it}}{1-(1-p)e^{it}}}$	${\frac {p}{1-(1-p)e^{it}}}$

Side by side comparison

Gamma
Probability density function
Cumulative distribution function
Parameters	k > 0 shape θ > 0 scale	α > 0 shape β > 0 rate
Support	$x\in (0,\infty )$	$x\in (0,\infty )$
PDF	${\frac {1}{\Gamma (k)\theta ^{k}}}x^{k-1}e^{-{\frac {x}{\theta }}}$	${\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}$
CDF	${\frac {1}{\Gamma (k)}}\gamma \left(k,{\frac {x}{\theta }}\right)$	${\frac {1}{\Gamma (\alpha )}}\gamma (\alpha ,\beta x)$
Mean	$\operatorname {E} [X]=k\theta$	$\operatorname {E} [X]={\frac {\alpha }{\beta }}$
Median	No simple closed form	No simple closed form
Mode	$(k-1)\theta {\text{ for }}k\geq 1$	${\frac {\alpha -1}{\beta }}{\text{ for }}\alpha \geq 1$
Variance	$\operatorname {Var} (X)=k\theta ^{2}$	$\operatorname {Var} (X)={\frac {\alpha }{\beta ^{2}}}$
Skewness	${\frac {2}{\sqrt {k}}}$	${\frac {2}{\sqrt {\alpha }}}$
Excess kurtosis	${\frac {6}{k}}$	${\frac {6}{\alpha }}$
Entropy	${\begin{aligned}k&+\ln \theta +\ln \Gamma (k)\\&+(1-k)\psi (k)\end{aligned}}$	${\begin{aligned}\alpha &-\ln \beta +\ln \Gamma (\alpha )\\&+(1-\alpha )\psi (\alpha )\end{aligned}}$
MGF	$(1-\theta t)^{-k}{\text{ for }}t<{\frac {1}{\theta }}$	$\left(1-{\frac {t}{\beta }}\right)^{-\alpha }{\text{ for }}t<\beta$
CF	$(1-\theta it)^{-k}$	$\left(1-{\frac {it}{\beta }}\right)^{-\alpha }$

Gamma
Probability density function
Cumulative distribution function
Parameters	k > 0 shape θ > 0 scale	α > 0 shape β > 0 rate
Support	$x\in (0,\infty )$	$x\in (0,\infty )$
PDF	${\frac {1}{\Gamma (k)\theta ^{k}}}x^{k-1}e^{-{\frac {x}{\theta }}}$	${\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}$
CDF	${\frac {1}{\Gamma (k)}}\gamma \left(k,{\frac {x}{\theta }}\right)$	${\frac {1}{\Gamma (\alpha )}}\gamma (\alpha ,\beta x)$
Mean	$\operatorname {E} [X]=k\theta$	$\operatorname {E} [X]={\frac {\alpha }{\beta }}$
Median	No simple closed form	No simple closed form
Mode	$(k-1)\theta {\text{ for }}k\geq 1$	${\frac {\alpha -1}{\beta }}{\text{ for }}\alpha \geq 1$
Variance	$\operatorname {Var} (X)=k\theta ^{2}$	$\operatorname {Var} (X)={\frac {\alpha }{\beta ^{2}}}$
Skewness	${\frac {2}{\sqrt {k}}}$	${\frac {2}{\sqrt {\alpha }}}$
Ex. kurtosis	${\frac {6}{k}}$	${\frac {6}{\alpha }}$
Entropy	${\begin{aligned}k&+\ln \theta +\ln \Gamma (k)\\&+(1-k)\psi (k)\end{aligned}}$	${\begin{aligned}\alpha &-\ln \beta +\ln \Gamma (\alpha )\\&+(1-\alpha )\psi (\alpha )\end{aligned}}$
MGF	$(1-\theta t)^{-k}{\text{ for }}t<{\frac {1}{\theta }}$	$\left(1-{\frac {t}{\beta }}\right)^{-\alpha }{\text{ for }}t<\beta$
CF	$(1-\theta it)^{-k}$	$\left(1-{\frac {it}{\beta }}\right)^{-\alpha }$