Template:Infobox probability distribution/testcases

Binomial distribution
	Probability mass functionProbability mass function for the binomial distribution
	Cumulative distribution functionCumulative distribution function for the binomial distribution
Notation	B(n,p)
Parameters	n∈{0,1,2,…} – number of trials; p∈[0,1] – success probability for each trial
Support	k∈{0,1,…,n} – number of successes
PMF	(nk)pk(1−p)n−k
CDF	I1−p(n−k,1+k)
Mean	np
Median	⌊np⌋ or ⌈np⌉
Mode	⌊(n+1)p⌋ or ⌈(n+1)p⌉−1
Variance	np(1−p)
Skewness	1−2pnp(1−p)
Excess kurtosis	1−6p(1−p)np(1−p)
Entropy	12log2(2πenp(1−p))+O(1n); in shannons. For nats, use the natural log in the log.
MGF	(1−p+pet)n
CF	(1−p+peit)n
PGF	G(z)=[(1−p)+pz]n
Fisher information	gn(p)=np(1−p); (for fixed n)

Normal Distribution
	Probability density functionProbability density function for the normal distribution; The red curve is the standard normal distribution
	Cumulative distribution functionCumulative distribution function for the normal distribution
Notation	𝒩(μ,σ2)
Parameters	μ∈ℝ = mean (location); σ2>0 = variance (squared scale)
Support	x∈ℝ
PDF	12πσ2e−(x−μ)22σ2
CDF	12[1+erf⁡(x−μσ2)]
Quantile	μ+σ2erf−1(2p−1)
Mean	μ
Median	μ
Mode	μ
Variance	σ2
MAD	2/πσ
Skewness	0
Ex. kurtosis	0
Entropy	12log⁡(2πeσ2)
MGF	exp⁡(μt+σ2t2/2)
CF	exp⁡(iμt−σ2t2/2)
Fisher information	ℐ(μ,σ)=(1/σ2002/σ2); ℐ(μ,σ2)=(1/σ2001/(2σ4))
Kullback-Leibler divergence	DKL(𝒩0‖𝒩1)=12{(σ0/σ1)2+(μ1−μ0)2σ12−1+2ln⁡σ1σ0}

Normal Distribution
	Probability density functionProbability density function for the normal distribution; The red curve is the standard normal distribution
	Cumulative distribution functionCumulative distribution function for the normal distribution
Notation	𝒩(μ,σ2)
Parameters	μ∈ℝ = mean (location); σ2>0 = variance (squared scale)
Support	x∈ℝ
PDF	12πσ2e−(x−μ)22σ2
CDF	12[1+erf⁡(x−μσ2)]
Quantile	μ+σ2erf−1(2p−1)
Mean	μ
Median	μ
Mode	μ
Variance	σ2
MAD	2/πσ
Skewness	0
Excess kurtosis	0
Entropy	12log⁡(2πeσ2)
MGF	exp⁡(μt+σ2t2/2)
CF	exp⁡(iμt−σ2t2/2)
Fisher information	ℐ(μ,σ)=(1/σ2002/σ2); ℐ(μ,σ2)=(1/σ2001/(2σ4))
Kullback–Leibler divergence	DKL(𝒩0‖𝒩1)=12{(σ0/σ1)2+(μ1−μ0)2σ12−1+2ln⁡σ1σ0}

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Normal distribution

Side by side comparison

Side by side comparison

Binomial distribution
Probability mass function Probability mass function for the binomial distribution
Cumulative distribution function Cumulative distribution function for the binomial distribution
Notation	$B (n, p)$
Parameters	$n \in {0, 1, 2, \dots}$ – number of trials $p \in [0, 1]$ – success probability for each trial
Support	$k \in {0, 1, \dots, n}$ – number of successes
PMF	$(\binom{n}{k}) p^{k} (1 - p)^{n - k}$
CDF	$I_{1 - p} (n - k, 1 + k)$
Mean	$n p$
Median	$⌊ n p ⌋$ or $⌈ n p ⌉$
Mode	$⌊ (n + 1) p ⌋$ or $⌈ (n + 1) p ⌉ - 1$
Variance	$n p (1 - p)$
Skewness	$\frac{1 - 2 p}{\sqrt{n p (1 - p)}}$
Ex. kurtosis	$\frac{1 - 6 p (1 - p)}{n p (1 - p)}$
Entropy	$\frac{1}{2} \log_{2} (2 π e n p (1 - p)) + O (\frac{1}{n})$ in shannons. For nats, use the natural log in the log.
MGF	$(1 - p + p e^{t})^{n}$
CF	$(1 - p + p e^{i t})^{n}$
PGF	$G (z) = [(1 - p) + p z]^{n}$
Fisher information	$g_{n} (p) = \frac{n}{p (1 - p)}$ (for fixed $n$ )

Side by side comparison

Geometric
Probability mass function File:Geometric pmf.svg
Cumulative distribution function File:Geometric cdf.svg
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	$⌈ \frac{- 1}{\log_{2} (1 - p)} ⌉$ (not unique if $- 1 / \log_{2} (1 - p)$ is an integer)	$⌈ \frac{- 1}{\log_{2} (1 - p)} ⌉ - 1$ (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	$\frac{- (1 - p) \log_{2} (1 - p) - p \log_{2} p}{p}$	$\frac{- (1 - p) \log_{2} (1 - p) - p \log_{2} p}{p}$
MGF	$\frac{p e^{t}}{1 - (1 - p) e^{t}},$ for $t < - \ln (1 - p)$	$\frac{p}{1 - (1 - p) e^{t}}$
CF	$\frac{p e^{i t}}{1 - (1 - p) e^{i t}}$	$\frac{p}{1 - (1 - p) e^{i t}}$

Geometric
Probability mass function File:Geometric pmf.svg
Cumulative distribution function File:Geometric cdf.svg
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	$⌈ \frac{- 1}{\log_{2} (1 - p)} ⌉$ (not unique if $- 1 / \log_{2} (1 - p)$ is an integer)	$⌈ \frac{- 1}{\log_{2} (1 - p)} ⌉ - 1$ (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	$\frac{- (1 - p) \log_{2} (1 - p) - p \log_{2} p}{p}$	$\frac{- (1 - p) \log_{2} (1 - p) - p \log_{2} p}{p}$
MGF	$\frac{p e^{t}}{1 - (1 - p) e^{t}},$ for $t < - \ln (1 - p)$	$\frac{p}{1 - (1 - p) e^{t}}$
CF	$\frac{p e^{i t}}{1 - (1 - p) e^{i t}}$	$\frac{p}{1 - (1 - p) e^{i t}}$

Side by side comparison

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

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