points in $Y$, then the map $\sigma$ reduces to a map $\varphi:G/K\times
Y\rightarrow X$. Suppose there is a $G$-invariant measure $d\mu$ on $G/K$, and suppose $\dim(G/K\times Y)=\dim X$, then it can be proved that the pull back measure $\varphi^*(p(x)dx)$ of the measure $p(x)dx$ is of the form $\varphi^*(p(x)dx)=d\mu d\nu$ for some measure $d\nu$ on $Y$, which is just the measure associated with the eigenvalue distribution. The measure $d\nu$ can be expressed as the form $d\nu(y)=\mathcal{P}(y)dy$ for some function $\mathcal{P}(y)$ on $Y$, which is just the joint density function. We write $\mathcal{P}(y)$ as the form $\mathcal{P}(y)=p(y)J(y)$, then under some orthogonality condition (that is $T_yY\perp
T_yO_y$ for almost all $y\in Y$), we can compute the factor $J(y)$ by the following formula $$\label{E:main}
J(y)=C|\det\Psi_y|,$$ where $C$ is a constant, which can also be computed explicitly. This formula is the main result of this paper, the density function $\mathcal{P}(y)$ and the eigenvalue distribution $d\nu$ are determined by it. Here the map $\Psi_y:{\mathfrak{l}}\rightarrow T_yO_y$ is defined by $$\Psi_y(\xi)=\frac{d}{dt}\Big|_{t=0}\hskip 0.1cm \sigma_{\exp
t\xi}(y),$$ where ${\mathfrak{l}}$ is a linear subspace of the Lie algebra ${\mathfrak{g}}$ of $G$ such that ${\mathfrak{g}}={\mathfrak{k}}\oplus{\mathfrak{l}}$, ${\mathfrak{k}}$ is the Lie algebra of $K$. We call the system $(G,\sigma,X,p(x)dx,Y,dy)$ a *generalized random matrix ensemble*. The 