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Gamma
Gamma
	Probability density function
	Cumulative distribution function
Parameters	k > 0 shape; θ > 0 scale;
Support	x∈(0,∞)
PDF	1Γ(k)θkxk−1e−xθ
CDF	1Γ(k)γ(k,xθ)
Mean	E⁡[X]=kθ
Median	No simple closed form
Mode	(k−1)θ for k≥1
Variance	Var⁡(X)=kθ2
Skewness	2k
Excess kurtosis	6k
Entropy	k+ln⁡θ+ln⁡Γ(k)+(1−k)ψ(k)
MGF	(1−θt)−k for t<1θ
CF	(1−θit)−k

Different texts adopt slightly different definitions for the negative binomial distribution. They can be distinguished by whether the support starts at k = 0 or at k = r, whether p denotes the probability of a success or of a failure, and whether r represents success or failure, so it is crucial to identify the specific parametrization used in any given text.
	Probability mass function; The orange line represents the mean, which is equal to 10 in each of these plots; the green line shows the standard deviation.
Notation	NB2(μ,ϕ)
Parameters	r > 0 — number of failures until the experiment is stopped (integer, but the definition can also be extended to reals); p ∈ (0,1) — success probability in each experiment (real)
Support	k ∈ { 0, 1, 2, 3, … } — number of successes
PMF	(k+r−1k)⋅(1−p)rpk, involving a binomial coefficient
CDF	1−Ip(k+1,r), the regularized incomplete beta function
Mean	μ
Mode	{⌊p(r−1)1−p⌋if r>10if r≤1
Variance	μ+μ2ϕ
Skewness	1+ppr
Excess kurtosis	6r+(1−p)2pr
MGF	(1−p1−pet)r for t<−log⁡p
CF	(1−p1−peit)r with t∈ℝ
PGF	(1−p1−pz)r for \|z\|<1p
Fisher information	r(1−p)2p

Different texts (and even different parts of this article) adopt slightly different definitions for the negative binomial distribution. They can be distinguished by whether the support starts at k = 0 or at k = r, whether p denotes the probability of a success or of a failure, and whether r represents success or failure, so identifying the specific parametrization used is crucial in any given text.
	Probability mass function; The orange line represents the mean, which is equal to 10 in each of these plots; the green line shows the standard deviation.
Notation	NB(r,p)
Parameters	r > 0 — number of successes until the experiment is stopped (integer, but the definition can also be extended to reals); p ∈ [0,1] — success probability in each experiment (real)
Support	k ∈ { 0, 1, 2, 3, … } — number of failures
PMF	k↦(k+r−1k)⋅(1−p)kpr, involving a binomial coefficient
CDF	k↦Ip(r,k+1), the regularized incomplete beta function
Mean	r(1−p)p
Mode	{⌊(r−1)(1−p)p⌋if r>10if r≤1
Variance	r(1−p)p2
Skewness	2−p(1−p)r
Excess kurtosis	6r+p2(1−p)r
MGF	(p1−(1−p)et)r for t<−log⁡(1−p)
CF	(p1−(1−p)eit)r with t∈ℝ
PGF	(p1−(1−p)z)r for \|z\|<11−p
Fisher information	ℐ(p,r)=(rp2(1−p)−1p−1p1−pr 3F2(1,1,1;2,r+1;1−p)), involving a generalized hypergeometric function
Method of moments	r=E[X]2V[X]−E[X] ; p=E[X]V[X]

The negative binomial distribution also arises as a continuous mixture of Poisson distributions (i.e. a compound probability distribution) where the mixing distribution of the Poisson rate is a gamma distribution. That is, we can view the negative binomial as a Poisson(λ) distribution, where λ is itself a random variable, distributed as a gamma distribution with shape = r and scale θ = p/(1 − p) or correspondingly rate β = (1 − p)/p.

1 2 DeGroot, Morris H. (1986). Probability and Statistics (Second ed.). Addison-Wesley. pp. 258–259. ISBN 0-201-11366-X. LCCN 84006269. OCLC 10605205.

[DeGrootNB-1] 1 2 DeGroot, Morris H. (1986). Probability and Statistics (Second ed.). Addison-Wesley. pp. 258–259. ISBN 0-201-11366-X. LCCN 84006269. OCLC 10605205.

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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	$\frac{1}{Γ (k) θ^{k}} x^{k - 1} e^{- \frac{x}{θ}}$	$\frac{β^{α}}{Γ (α)} x^{α - 1} e^{- β x}$
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	$\begin{matrix} k & + \ln θ + \ln Γ (k) \\ + (1 - k) ψ (k) \end{matrix}$	$\begin{matrix} α & - \ln β + \ln Γ (α) \\ + (1 - α) ψ (α) \end{matrix}$
MGF	$(1 - θ t)^{- k} for t < \frac{1}{θ}$	${(1 - \frac{t}{β})}^{- α} for t < β$
CF	$(1 - θ i t)^{- k}$	${(1 - \frac{i t}{β})}^{- α}$