There is a function $y$ defined
$$y=\exp(-\boldsymbol{\alpha}'\mathbf{b})\:\:;\:\:\:\:y\in(0,\infty)$$
where $\boldsymbol{\alpha}$ is a vector of random variables and $\mathbf{b}$ is a vector of non-random parameters.
So, the expected value of $\ln y$ is just
$$E[\ln y] = -E[\boldsymbol{\alpha}]'\mathbf{b}$$
My question is, how to go about finding a computable expression for $E[y]$ in terms of the means (and maybe standard deviations?) of $\boldsymbol{\alpha}$?
It seems like the law of the unconscious statistician should be of some use here. But I have been unable to make use of it.
Also, if it is known that $y$ is lognormally distributed ($\ln y \sim N(\mu,\sigma^2)$), does this facilitate things?