Let $X\in\mathbb{R}^{m\times n}$, where $m<n$, be a random matrix where the rows $x_i$ ($i=1,...,m$) are sampled i.i.d. from Gaussian distribution with mean $0$ and covariance $\Sigma$, i.e. $x_i\sim N(0,\Sigma)$.

How to calculate the expected value $\mathbb{E}[X^{+}X]$ where $X^{+}$ is the Moore–Penrose inverse of $X$ ?

Note: For the special case where $\Sigma=I$ it is easy to see that $\mathbb{E}[X^{+}X]=\frac{m}{n}I$.

Thank you.