# Expected value of trace of matrix inverse

Given a $N\times K$ matrix $A$ of full rank with $K < N$, a diagonal matrix $D$ and knowing that $E[D]=bI_N$, where $E[\cdot]$ is the expected value and $I_N$ is the $N\times N$ identity matrix and $b>0$ is a known scalar,

how can I compute (or bound): $E[\operatorname{trace}((A^H D A)^{-1} A^HA)]$

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There is information missing in this question related to the assumptions of the matrix $D$. Consider for example the case of $D$=0 with probability 0.5, and $D=2bI_N$ with probability 0.5. This satisfies the assumption $E[D]=bI_N$, but the matrix inverse is undefined with probability 0.5.