Expected Fourier weight after a random restriction (${\mathbb{E}}_{\rho }{W}_{>s}(f_{\rho })={\sum }_S|\hat{f}(S){|}^2\mathbb{P}(\mathrm{Bin}(|S|,p)>s)$)

For a random restriction, first fix the live set $J$ and average over the unbiased assignment $a$ outside it. The coefficient at $U\subset J$ is ${\sum }_{B\subset J^c}\hat{f}(U\cup B){\chi }_B(a)$. Orthogonality eliminates cross terms in its squared expectation. Averaging over $J$ gives the displayed identity. The original Fourier weight is weighted by a binomial distribution survival probability, not simply preserved.