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.

**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
$$\frac{\mathbb{E}\langle\mathbf{X},\mathbf{v}\rangle}{\mathbb{E}\,{\sum_{\!i=1}^{\!n}\!X_i}}>\frac{1}{n}\mathbb{E}\left[\frac{\langle\mathbf{X},\mathbf{v}\rangle}{{\sum_{\!i=1}^{\!n}\!X_i}}\right]\ ?$$

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