We are given a vector $n$-dimensional random vector $\mathbf{X}$ whose components are the Bernoulli random variables $X\_1, X\_2, \ldots X\_n$, such that the probability $\mathbb{P}(X\_1=X\_2=\ldots=X\_n=0)$ is null.




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**Question:** For $n>1$, is there any joint distribution of $\mathbf{X}$’s components and a vector $\mathbf{v}\in [0,1]^n$ such that
$$\frac{\mathbb{E}\langle\mathbf{X},\mathbf{v}\rangle}{\mathbb{E}\,{\sum\_{\!i=1}^{\!n}\!X\_i}}>\frac{1}{n}\mathbb{E}\left[\frac{\langle\mathbf{X},\mathbf{v}\rangle}{{\sum\_{\!i=1}^{\!n}\!X\_i}}\right]\ ?$$



On the mean value taken by Bernoulli random variables with joint distribution constraints
$\newcommand\X{\mathbf X}\newcommand\v{\mathbf v}$


Indeed, $1\le\sum\_{i=1}^n X\_i\le n$ and hence
$$n\sum\_{i=1}^n X\_i\ge n=En\ge E\sum\_{i=1}^n X\_i,$$
so that
$$\frac1n\,E\frac{\langle\X,\v\rangle}{\sum\_{i=1}^n X\_i}
=E\frac{\langle\X,\v\rangle}{n\sum\_{i=1}^n X\_i}\le 
E\frac{\langle\X,\v\rangle}{E\sum\_{i=1}^n X\_i}
=\frac{E\langle\X,\v\rangle}{E\sum\_{i=1}^n X\_i},$$
as claimed. $\quad\Box$


