# Non-linear transformation of univariate normal variables: $y_i=\frac{e^{x_i}}{\sum_{j=1}^{K}{e^{x_j}}}$

Let $\{x_i\in\mathbb{R}\}_{i=1}^{K}$ be a set of random univariate variables, each of which is distributed normally as $x_i\sim\mathcal{N}(\mu_{x,i},\sigma_{x,i}^2)$. Note that $x_i$'s are pairwise independent.

Also, let $\{y_i\in\mathbb{R}\}_{i=1}^{K}$ be the set of random variables, where each variable $y_i$ is given as the following non-linear transformation of $x_i$: $$y_i=\frac{e^{x_i}}{\sum_{j=1}^{K}{e^{x_j}}}.$$ How could we find $y_i$'s mean $\mu_{y,i}$ and variance $\sigma_{y,i}^2$?

• Nonnegative random variables distributed normally? This would be a surprise... – Did Oct 20 '16 at 11:51
• @Did correct, it's obviously not normal (I edit my question). But how could we compute $y_i$'s mean and variance? – nullgeppetto Oct 20 '16 at 11:57
• Without assuming joint normality, and for unspecified parameters $\mu_i$ and $\sigma^2_i$, I doubt there is a theoretical answer. For given values of the parameters, simulations might be an option. – Did Oct 20 '16 at 12:05
• OK, you did restrict the setting by assuming joint normality but you widened it by stopping to (implictely) assume independence. Anyway, even with joint normality and independence, "for unspecified parameters μi and σ2i, I doubt there is a theoretical answer. For given values of the parameters, simulations might be an option." – Did Oct 20 '16 at 12:42
• Specifications of this problem seem to be in flux. If this question is motivated by a practical application, you might want to look at Wikipedia on 'lognormal distributions' to see if anything there seems useful. – BruceET Oct 20 '16 at 16:03

You can use Unscented Transformation to find mean and covariance. Let $y = f(x)$ be a linear or nonlinear mapping from $x$ to $y$. Given that you know the mean and covariance of $x$, you can find mean and covariance of $y$ by generating sigma points corresponding to $x$.