Problem Definition: Suppose that a random variable of multivariate Gaussian distribution $X \sim N(\Sigma,\mu)$, $\Sigma$ is the covariance matrix, and $\mu$ is the mean. For each $x_i$ from $X$, $x_i \sim N(\sigma^2,\nu)$, where $\sigma^2$ is the variance and $\nu$ is the mean of $x_i$. So we can get the Expection $E[e^{x_i}] = e^{\nu + \frac{\sigma^2}{2}}$, because $e^{x_i}$ follows a log normal distribution.

Question: Randomly select some $x_i$ from $X$, what is the Expectation $E[\frac{e^{x_i}}{\sum_{(i,...,j)}e^{x_j}}]$?

For example: we pick up $x_1$, $x_4$, $x_9$, $x_{18}$, what is the $E[\frac{e^{x_1}}{e^{x_1}+e^{x_4}+e^{x_9}+e^{x_{18}}}]$, or $E[\frac{e^{x_9}}{e^{x_1}+e^{x_4}+e^{x_9}+e^{x_{18}}}]$, or like this.

Anyone can help me? Thanks a lot!

  • $\begingroup$ Your example is inconsistent with your question: different functional forms -- exponents or no exponents? $\endgroup$ – wolfies Apr 30 '14 at 18:26
  • $\begingroup$ Sorry, i mean 'exponents'. i have modified the example. Do you have any idea about this problem? $\endgroup$ – Double Gray May 22 '14 at 8:06

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