---
title: Bivariate Gamma Distribution Overview
url: https://www.emergentmind.com/topics/bi-variate-gamma-distribution
type: topic
---

# Bivariate Gamma Distribution Overview

The bi-variate gamma distribution, also frequently referred to as the two-variate (non-central) multivariate gamma law, generalizes the univariate gamma distribution to two dependent, non-negative random variables. The construction is naturally linked to the diagonal elements of non-central Wishart matrices and is parameterized by a positive shape parameter, a positive-definite scale/covariance matrix, and a symmetric non-centrality matrix. This distribution is of fundamental importance both as a mathematical object and as a tool in multivariate statistical analysis, distributional theory, and the study of correlated gamma-type random variables [1604.06906].

## 1. Definition and Parameterization

Let $\vec{X} = (X_1, X_2)^{\prime}$ be a random vector characterized by:

- **Shape parameter**: $\alpha > 0$, sometimes expressed in the Wishart context as degrees of freedom $\nu = 2\alpha$.
- **Scale (covariance) matrix**: $\Sigma = [\sigma_{ij}]_{2\times2} > 0$. It admits the factorization $\Sigma = W^{-2} R W^{-2}$, where $W = \operatorname{diag}(w_1, w_2)$ with $w_i = \sigma_{ii}^{-1/2}$, and $R = (r_{ij})$ is a correlation matrix such that $R_{ii}=1$, $R_{12}=R_{21}=\rho$, $|\rho|<1$.
- **Non-centrality matrix**: $\Delta = [\delta_{ij}]_{2\times2} \geq 0$, symmetric positive semi-definite, with frequent consideration of rank-one forms $\Delta = d d^\prime$ for some $d = (d_1, d_2)^\prime$.

The Laplace transform of the joint distribution is given by:
\[
\mathcal{L}_{X}(t_1, t_2) = E\bigl[e^{-t_1 X_1 - t_2 X_2}\bigr] = |I + R T|^{-\alpha} \exp\left( -\mathrm{tr}\left[ T (I + R T)^{-1} \Delta \right]\right)
\]
where $T = \operatorname{diag}(t_1, t_2) \geq 0$. This characterizes the distribution completely, extending the univariate gamma and non-central chi-square families [1604.06906].

## 2. Series Expansions for the Joint Density and Distribution Functions

For the bivariate gamma law, the CDF and PDF admit absolutely convergent double-series expansions. Introducing auxiliary quantities:
- $W = \operatorname{diag}(w_1, w_2)$,
- $B = W R W - I$,
- $D = W^{-1} \Sigma^{-1} \Delta \Sigma^{-1} W^{-1}$,
- $g_\alpha(x) = x^{\alpha-1} e^{-x}/\Gamma(\alpha)$ (central gamma density),
- $P(\alpha, x) = \gamma(\alpha, x)/\Gamma(\alpha)$ (regularized lower incomplete gamma),

the joint CDF is
\[
G(x_1, x_2) = \exp\!\left(-\operatorname{tr}\left(\Delta \Sigma^{-1}\right)\right) \sum_{n_1=0}^\infty \sum_{n_2=0}^\infty \frac{d(n_1, n_2)}{n_1! n_2!} \;
P(\alpha+n_1; w_1^2 x_1) \,
P(\alpha+n_2; w_2^2 x_2)
\]
where, for $p=2$,
\[
d(n_1, n_2) = 
\begin{cases}
\frac{(B_{12})^{n_1+n_2} (n_1 + n_2)!}{n_1! \, n_2!}, & n_1 + n_2 \text{ even}, \\
0, & n_1 + n_2 \text{ odd}.
\end{cases}
\]
Differentiating term-by-term yields the joint PDF:
\[
f(x_1, x_2) = \exp\left(-\operatorname{tr}(\Delta \Sigma^{-1})\right)
\sum_{n_1, n_2 \geq 0} \frac{d(n_1, n_2)}{n_1! n_2!}
g_{\alpha + n_1}(w_1^2 x_1) \,
g_{\alpha + n_2}(w_2^2 x_2)
\]
The structure of the series reflects how correlation and non-centrality couple the marginal gamma laws [1604.06906].

## 3. Integral Representations

The cumulative distribution function also admits a single-integral (Fourier-type) representation by virtue of every $2 \times 2$ correlation matrix being "one-factorial." Define:
- $\rho = \sigma_{12} / (\sigma_{11}\sigma_{22})^{1/2}$,
- $y = \rho e^{i\varphi}$, $\bar y = \rho e^{-i\varphi}$,
- $d_{ij} = [W^{-1} \Sigma^{-1} \Delta \Sigma^{-1} W^{-1}]_{ij}$,
- $\Lambda = \operatorname{tr}(\Delta \Sigma^{-1})$,
- $\widetilde{P}(\alpha; u, v) = \exp(-v) \sum_{k=0}^\infty \frac{v^k}{k!} P(\alpha+k; u)$,

Then, the CDF has the form
\[
G(x_1, x_2) = (1-\rho^2)^{\alpha} e^{-\Lambda} \frac{1}{2\pi}
\int_0^{2\pi} \exp\left(d_{11} y + d_{22} \bar y\right)
(1-\rho^2 y)^{-\alpha}
\,\widetilde{P}\left(\alpha; w_2^2 x_2 (1-\rho^2 \bar y), d_{22} \bar y\right)
\, d\varphi
\]
This integral is highly tractable, converges rapidly, and reveals the analytic structure imposed by both the correlation coefficient and the non-centrality matrix. Differentiation under the integral sign provides an analogous formula for the joint PDF [1604.06906].

## 4. Influence of Non-Centrality and Correlation

The non-centrality matrix $\Delta$ appears in the formulae solely through the scalar $\Lambda = \operatorname{tr}(\Sigma^{-1}\Delta)$ and the entries $d_{ij}$ of $D = W^{-1} \Sigma^{-1} \Delta \Sigma^{-1} W^{-1}$. The correlation $\rho$ influences (i) the prefactor $(1-\rho^2)^{\alpha}$, (ii) the power-law and exponential terms in the integrand (via $y = \rho e^{i\varphi}$), and (iii) the scale factors $w_i = \sigma_{ii}^{-1/2}$. 

In the case of rank-one non-centrality, $\Delta = dd'$, the exponent in the integrand specializes to $d_{11}y + d_{22}\bar y + 2\,\Re\{d_{12}y\}$, further simplifying computational realization. This form offers insight into the coupling mechanism between marginals as driven jointly by non-centrality and correlation, particularly for practical simulation or evaluation [1604.06906].

## 5. Special Cases and Computational Considerations

Distinct important specializations include:
- **Central case ($\Delta = 0$)**: $\Lambda=0$, $d_{ij}=0$, eliminating the exponential term; the result reduces to the Appell $F_1$ (or Gauss hypergeometric) representation for the central bivariate gamma/chi-square law.
- **Equal-scale case ($\sigma_{11} = \sigma_{22}$)**: $w_1 = w_2$, yielding a symmetric appearance of $y$ in $x_1$ and $x_2$.
- **Independent marginals ($\rho=0$)**: The integral collapses to the product $P(\alpha; w_1^2 x_1) P(\alpha; w_2^2 x_2)$, i.e., independence in the joint law.
- **Rank-one non-centrality ($\Delta = dd'$)**: The CDF admits the simpler single-sum expansion:
  \[
  G(x_1, x_2) = e^{-\|W^{-1}d\|^2} \sum_{k=0}^{\infty}
  \frac{\left(\|W^{-1}d\|^2\right)^k}{k!}
  P(\alpha+k; w_1^2 x_1)
  P(\alpha+k; w_2^2 x_2)
  \]

In practical calculations, the integral over $\varphi \in [0, 2\pi]$ is truncated or mapped to $[-\pi, \pi]$ and computed using Gaussian or Fourier-type quadrature; convergence is rapid, since all singularities are exterior to the integration path $|y|=|\rho|<1$. This structure affords both (i) a series expansion in non-centrality, positive and real, and (ii) a compact, analytically transparent, single-integral representation [1604.06906].

## 6. Connections to Wishart Distributions and Broader Significance

The bi-variate gamma law emerges as the diagonal marginal distribution of a $2 \times \nu$ non-central Wishart matrix $W_2(\nu, \Sigma, \Delta)$ with $\nu = 2\alpha$. The general framework extends to higher-dimensional (p-variate) settings, with $p-1$-dimensional integral representations for the CDF. For $p=2$, these reduce to the explicit, computationally tractable forms described. This constructs a systematic link between marginal gamma-type variables with arbitrary non-centrality and correlation, bridging chi-square, Wishart, and multivariate gamma-analytic methods. Alternative formulas exist in the case of "one-factorial" correlation matrices, leading to further simplifications for $p=3$ and certain structured correlation models.

The principal results and representations outlined here, as developed in the work of Royen [1604.06906], are foundational for computation, theory, and application of correlated gamma variates in multivariate statistics and probabilistic analysis.

Source: https://www.emergentmind.com/topics/bi-variate-gamma-distribution