I am working on the problem related to expectation of convex function of a Poisson Binomial variable. Any help would be appreciated.
First, there is a Poisson Binomial distributed random variable $B$, i.e., $B = \sum_{i=1}^K x_i$, such that $x_i$ is Bernoulli distributed with parameter $P_i$. There is no necessity for variables $x_i$ to be identically distributed but they are independent.
Second, there is an objective function, $O = \sum_{k=0}^K P\{B=k\}\varphi(k)$. We can see that it is the expectation of $\varphi(B)$, where the function $\varphi$ is assumed to be a monotonically increasing and strictly convex function.
Then, we want to compare the objective functions with two different distribution of $x_i$. Assume that for each variable $x_i$, $P_{i,1}\le P_{i,2}$. Could we conclude that $O_1 \le O_2$ in general or under some special cases for the defined function $\varphi$?
What I have learned is that for variable $B$, $\operatorname{E}(B)=\sum_{i=1}^K P_i$ and the variance is $\operatorname{Var}(B) = \sum_{i=1}^K P_i(1-P_i)$. Intuitively, we know that the variance is increasing and strictly convex function of $B$. Since $\frac{\partial \operatorname{Var}(B)}{\partial P_i}=1-2P_i$, we coould conclude that it is monotonically increasing only when $P_i \le 1/2$. However, I need to demonstrate it without any other constraints on the distribution $P_i$.
Any other results under any specific convex function or more general solution would be appreciated. Thanks a lot!