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Recently Referred Formulae

y(t) &=& y(0)   (\mu - \frac{1}{2}\sigma^2)t   \sigma W(t) \\
x(t) &=& x(0)\exp\left( (\mu - \frac{1}{2}\sigma^2)t   \sigma W(t) \right)
{\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
X_N^{(1)} &=& (1-\lambda)X_0   \lambda R_1 R_2 \cdots R_N X_0 \\
          &=& (1-\lambda)X_0   \lambda X_0 \prod_{i=1}^N R_i
{\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 E}\left[ {\cal R}_i {\cal R}_j \right] = \mu^{|i-j|}(\mu^2   \sigma^2)^{\min(N-i 1, N-j 1)}