Suppose that each entry in $n$ by $p$ matrix $X$ has standard normal distribution $\mathcal{N}(0,1)$. I am interested in finding the proof that
$\mathbb{E}(X(X'X)^{-1}X') = \frac pn \cdot I_n$,
which is my guess coming from simulations. I have tried to attack the problem using Wishard distributions but without results. I showed that:
$X(X'X)^{-1}X'$ is symmetric
trace of $X(X'X)^{-1}X'$ is equal to $p$
$X(X'X)^{-1}X'$ is idempotent matrix
I would prefer to write a comment as I cannot solve your problem but I do not have enough reputation and want to add one point that could be important: $$X(X'X)^{-1}X'=XX^+X'^+X'=PQ$$ were $P$ is the orthogonal projection onto the range of $X$ and Q is the orthogonal projection onto the range of $X'$ and $(\cdot)^+$ denotes the Moore-Penrose Pseudoinverse. Handling stochastic input parameters makes everything more complicated but I suppose that the range of $X$ should not be affected a.s. by drawing normal distributed variables.