Share The Formula You Found

+Label +Description

Recently Referred Formulae

X_N^{(2)} &=& \frac{X_0}{N}R_1 R_2 \cdots R_N + \frac{X_0}{N} R_2 R_3 \cdots R_N + \cdots + \frac{X_0}{N}R_N \\
          &=& \frac{X_0}{N} (R_1 \cdots R_N + R_2 \cdots R_N + \cdots + R_N) \\
          &=& \frac{X_0}{N} \sum_{i=1}^N \prod_{j=i}^N R_j \\
          &\equiv& \frac{X_0}{N} \sum_{i=1}^N {\cal R}_i
{\rm E}\left[ {\cal R}_i {\cal R}_j \right] = {\rm E}\left[ R_i \cdots R_N \times R_j \cdots R_N \right]
{\rm V}[R_1 R_2 \cdots R_N] &=& {\rm E}[R_1^2 R_2^2 \cdots R_N^2] - ({\rm E}[R_1 R_2 \cdots R_N])^2 \\
 &=& {\rm E}[R_1^2] \cdots {\rm E}[R_N^2] - \mu^{2N}
{\rm E}[X_N^{(2)}] &=& \frac{X_0}{N}\left( \mu^N + \mu^{N-1} + \cdots + \mu \right) \\
 &=& \frac{X_0}{N}\sum_{i=1}^N \mu^i
{\rm V}[X_N^{(2)}] &=& \frac{X_0^2}{N^2} {\rm V}[\sum_{i=1}^N {\cal R}_i] \\
 &=& \frac{X_0^2}{N^2} \left\{ {\rm E}\left[ \left( \sum_{i=1}^N {\cal R}_i \right)^2 \right] - \left( {\rm E}\left[ \sum_{i=1}^N {\cal R}_i \right] \right)^2 \right\}