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Let X be a chi-square variable with two degrees of freedom. Let A and B be to arbitrary constants, with $A>B>0$.

I need the variance of $Y=\log(1+AX)-\log(1+BX).$

The mean is, maybe not simple, but doable, since it is the ergodic channel capacity of a Reyleigh fading channel, well known in communication theory, and is given by a sum of Exponential integrals, unless I recall incorrectly.

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  • $\begingroup$ Indeed, $\int_0^\infty e^{-X}Y^2\,dX$ does not seem to have a closed-form expression. $\endgroup$ Mar 18, 2016 at 20:23

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