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{\rm V}[R_1 R_2 \cdots R_N] &=& (\sigma^2   \mu^2)^N - \mu^{2N} \\
 &=& \sum_{i=0}^N {}_N C_i \sigma^{2i}\mu^{2(N-i)} - \mu^{2N} \\
 &=& \sum_{i=1}^N {}_N C_i \sigma^{2i}\mu^{2(N-i)}
\mathrm{p}_{ij} = \left\{ \begin{array}{ll}
i^2/N^2 & j=i-1 \\
2i(N-i)/N^2 & j=i \\
(N-i)^2/N^2 & j=i 1 \\
0 & \mathrm{otherwise}
\end{array} \right.
&& {\rm scan}\;\;(\oplus)\;\;[a_0, \cdots , a_n] \\
&& = [a_0, a_0 \oplus a_1, \cdots , a_0 \oplus \cdots \oplus a_n] \\
&& \equiv [b_0, \cdots, b_n]
\left( {\rm E}\left[ \sum_{i=1}^N {\cal R}_i \right] \right)^2 &=& (\mu^N   \mu^{N-1}   \cdots   \mu)^2 \\
 &=& \left( \sum_{i=1}^N \mu^i \right)^2
{\rm E}[X_n^{(1)}] &=& (1-\lambda)X_0   \lambda {\rm E}[R_1] \cdots {\rm E}[R_N] X_0 \\
 &=& (1-\lambda)X_0   \lambda \mu^N X_0