ThmDex – An index of mathematical definitions, results, and conjectures.
Result R5679 on D207: Bernoulli random boolean number
Given convex coefficient parameters, equal allocation maximizes uncorrelated finite Bernoulli product
Formulation 0
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}
Proofs
Proof 0
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}
By uncorrelatedness and by result R5284: Expectation of a Bernoulli random boolean number, we have \begin{equation} \mathbb{E} \prod_{n = 1}^N X_n = \prod_{n = 1}^N \mathbb{E} X_n = \prod_{n = 1}^N \theta_n \end{equation} Since also \begin{equation} N^{- N} = \left( \frac{\sum_{n = 1}^N \theta_n}{N} \right)^N \end{equation} then this result is a special case of R5182: Tight upper bound to a finite product of unsigned real numbers. $\square$