  Let $s\_2(x)$ denote the number of digits "$1$" in a binary expansion of any
$x \in \mathbb{N}$. We study the mean distribution $\mu\_a$ of the quantity
$s\_2(x+a)-s\_2(x)$ for a fixed positive integer $a$.It is shown that solutions
of the equation$$ s\_2(x+a)-s\_2(x)= d $$are uniquely identified by a finite
set of prefixes in $\{0,1\}^*$, and that the probability distribution of
differences $d$ is given by an infinite product of matrices whose coefficients
are operators of $l^1(\mathbb{Z})$.Then, denoting by $l(a)$ the number of
patterns "$01$" in the binary expansion of $a$, we give the asymptotic
behaviour of this probability distribution as $l(a)$ goes to infinity as well
as estimates of the variance of the probability measure $\mu\_a$
