On a property of the soft bit density associated with loglikelihood ratios of a binary AWGN channel

We derive a relation between the unconditional and conditional (conditioned on the input bit X to be either 0 or 1) densities of the expected bit value, the so-called`soft bit' ¿, given the log-likelihood ratios L(X) at the output of an AWGN channel. Both quantities are connected via the relation ¿ = tanh(L/2). Denoting the unimodal, conditional soft bit density as f
_¿lx,uni
(¿|x) and the bimodal unconditional density as f
_¿,bi
(¿) we prove the following property: the mean of the conditional soft bit distribution is equal to the variance of the unconditional soft bit ¿
_¿,bi

²
, i.e. ¿
_¿|X,uni
= ¿
_¿,bi

²
. This property is rooted in the consistency property of the associated L-value density.