I am looking for any way to prove the equivalence of these two maximization problems $$ \mathop{\mathrm{argmax}}_{\mathbf{\lambda}} \mathbb{E}_{X}\Big[\sum_n^N a_n\log_2(X(\mathbf{k};\theta)))\Big] \triangleq \mathop{\mathrm{argmax}}_{\mathbf{\lambda}} \sum_n^N a_n\log_2(\mathbb{E}_{X}[X(\mathbf{k};\theta)]) $$ where $a_n$ are always positive and $X$ is a r.v. which stands as a non-negative composition of random variables. The vector $\mathbf{k}$ has dimension $N$ and is made up by Poisson random variables the n-th of those has mean $\mathbb{E}[k_n]=\lambda_n$. $\theta$ is the set of random variables that compose $X$ but do not affect the realization of the other terms of the sum.
Please, note the second term of the equivalence is the application of Jensen's inequality to the left-hand side maximization. This leads to the fact that the objective's maximum can't be the same for the two problems. If the maximizers of the two problems are exactly the same optimum feasible set I have solve my issue.
Any hint is highly appreciated!