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. $\square$
