Let $X(N,N)$ be Wishart matrix with rank(X)=K in order to estimate the expectation of the trace of the square root of X i.e $X^{1/2}$ I want to know if is possible to use the unordered Wishart distribution function to estimate this value?


\begin{align}
E[trace(\sqrt X )]=?
\end{align}



expectation of the trace of the square root of wishart matrix
Since the trace is invariant under unitary transformations, you can work in a basis where $X$ is diagonal, with nonzero elements $x\_n$, $n=1,2,\ldots K$ on the diagonal; denote by $P(x)$ their marginal distribution; then 
$$\mathbb{E}[{\rm tr}\, \sqrt X]=\mathbb{E}\left[\sum\_{n=1}^K\sqrt{x\_n}\right]=K\int P(x)\sqrt{x}\,dx.$$
For $K\gg 1$ you can use the [Marcenko-Pastur distribution](https://en.wikipedia.org/wiki/Marchenko%E2%80%93Pastur_distribution) for $P(x)$.


