In a confined NIO space, there are nnn NIO particles, the iii-th of which has aia_iai joule energy. The NIO particles are very special as they keep colliding with each other randomly. When one particle carrying energy aaa joule collides with another particle carrying energy bbb joule, they will be annihilated and produce two new particles carrying the energy of a AND ba\ \texttt{AND}\ ba AND b and a OR ba\ \texttt{OR}\ ba OR b respectively. Here AND\texttt{AND}AND and OR\texttt{OR}OR mean bitwise AND and OR operation respectively.
The variance of the energy of these particles is obviously not decreasing, but unfortunately, the space here is too small for the author to write down his proof. After enough time the variance of the energy of these particles converges to a stable value. Can you find this value?
The variance of
nnn numbers is defined as follows.
σ2=1n∑i=1n(xi−μ)2where μ=1n∑i=1nxi\begin{aligned}
\sigma^2 &= \frac{1}{n}\sum\limits_{i=1}^{n}(x_i - \mu)^{2} \\
\text{where}\ \mu &= \frac{1}{n}\sum\limits_{i=1}^{n}x_i \\
\end{aligned}σ2where μ=n1i=1∑n(xi−μ)2=n1i=1∑nxi