Random restriction of a Boolean function (${\mathcal{R}}_p$)

A $p$-random restriction independently leaves each coordinate live with probability $p$, otherwise fixing it to zero or one with probability $(1-p)/2$ each. Composing independent restrictions with live probabilities $p_1,\dots ,p_d$ produces the same distribution with live probability ${\prod }_j{p}_j$. Fixed values stay unbiased.