  Exponential family plays an important role in information geometry. In
arXiv:1811.01394, we introduced a method to construct an exponential family
$\mathcal{P}=\{p_\theta\}_{\theta\in\Theta}$ on a homogeneous space $G/H$ from
a pair $(V,v_0)$. Here $V$ is a representation of $G$ and $v_0$ is an $H$-fixed
vector in $V$. Then the following questions naturally arise: (Q1) when is the
correspondence $\theta\mapsto p_\theta$ injective? (Q2) when do distinct pairs
$(V,v_0)$ and $(V',v_0')$ generate the same family? In this paper, we answer
these two questions (Theorems 1 and 2). Moreover, in Section 3, we consider the
case $(G,H)=(\mathbb{R}_{>0}, \{1\})$ with a certain representation on
$\mathbb{R}^2$. Then we see the family obtained by our method is essentially
generalized inverse Gaussian distribution (GIG).
