I am trying to find an upperbound for $$E_I\left[\sum_{k=I}^n a_k^2 - \sum_{k=I}^n b_k^2\right],$$ where the expectation is taken over the random variable $I$. The tuples $a$ and $b$ are real and $I$ may take values on $\{1,...,n\}$.
I checked Jensen's inequalites but without success. Any help would be appreciated.