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Recently Referred Formulae

{\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)}
{\rm V}[X_N^{(2)}] &=& \frac{X_0^2}{N^2} \left\{ 2\sum_{j>i}\mu^{j-i}(\mu^2 +\sigma^2)^{N-j+1} + \sum_{i=1}^N (\mu^2 +\sigma^2)^i - \left( \sum_{i=1}^{N}\mu^i \right)^2 \right\} \\
 &\equiv& \frac{X_0^2}{N^2} \left\{ 2\sum_{j>i}\mu^{j-i}\nu^{N-j+1} + \sum_{i=1}^N \nu^i - \left( \sum_{i=1}^{N}\mu^i \right)^2 \right\}
{\rm V}[X_N^{(1)}] &=& \lambda^2 X_0^2 \sum_{i=1}^N {}_N C_i \sigma^{2i}\mu^{2(N-i)}
{\rm V}[X_N^{(2)}] &=& \frac{X_0^2}{N^2} \left\{ 2\sum_{j>i} \mu^{N - j + 1}\mu^{N - i + 1} + \sum_{i=1}^N \mu^i -\left( \sum_{i=1}^N \mu^i \right)^2 \right\} \\
 &=& \frac{X_0^2}{N^2} \left\{ 2\sum_{j<i} \mu^{i}\mu^{j} + \sum_{i=1}^N \mu^i -\left( \sum_{i=1}^N \mu^i \right)^2 \right\} \\
 &=& 0
{\rm E}\left[ {\cal R}_i {\cal R}_j \right] = \mu^{|i-j|}(\mu^2 + \sigma^2)^{\min(N-i+1, N-j+1)}