Let $X_1, \, \ldots, \, X_N \in \text{Random} \{ 0, 1 \}$ each be a D3628: Random Boolean number such that

(i)	\begin{equation} X_1 \overset{d}{=} \text{Bernoulli} (\theta_1), \quad \ldots, \quad X_N \overset{d}{=} \text{Bernoulli} (\theta_N) \end{equation}
(ii)	$X_1, \, \ldots, \, X_N$ is an D3842: Uncorrelated random collection
(iii)	\begin{equation} \sum_{n = 1}^N \theta_n = 1 \end{equation}

Then

(1)	\begin{equation} \mathbb{E} \prod_{n = 1}^N X_n \leq N^{-N} \end{equation}
(2)	\begin{equation} \mathbb{E} \prod_{n = 1}^N X_n = N^{-N} \quad \iff \quad \theta_1 = \theta_2 = \cdots = \theta_N = \frac{1}{N} \end{equation}