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In Sabot and Tarr\`es (2015), the authors have explicitly computed theintegral $$STZ_n=\int \exp( -\langle x,y\rangle)(\det M_x)^{-1/2}dx$$ where$M_x$ is a symmetric matrix of order $n$ with fixed non positive off-diagonalcoefficients and with diagonal $(2x_1,\ldots,2x_n)$. The domain of integrationis the part of $\mathbb{R}^n$ for which $M_x$ is positive definite. Wecalculate more generally for $ b_1\geq 0,\ldots b_n\geq 0$ the integral$$GSTZ_n=\int \exp \left(-\langlex,y\rangle-\frac{1}{2}b^*M_x^{-1}b\right)(\det M_x)^{-1/2}dx,$$ we show that itleads to a natural family of distributions in $\mathbb{R}^n$, called the$GSTZ_n$ probability laws. This family is stable by marginalization and byconditioning, and it has number of properties which are multivariate versionsof familiar properties of univariate reciprocal inverse Gaussian distribution.We also show that if the graph with the set of vertices $V=\{1,\ldots,n\}$ andthe set $E$ of edges $\{i,j\}'$ s of non zero entries of $M_x$ is a tree, thenthe integral $$\int \exp( -\langle x,y\rangle)(\det M_x)^{q-1}dx$$ where $q>0,$is computable in terms of the MacDonald function $K_q.$

Multivariate reciprocal inverse Gaussian distributions from the Sabot -Tarr\`es -Zeng integral