Discrete derivative of a Boolean function ($D_{i}f$)

On the unbiased cube, the discrete derivative isFor the p-biased product measure, the normalization factor is $\sigma /2$.

The influence of coordinate $i$ is ${\mathrm{Inf}}_i(f)=\Vert D_{i}f{\Vert }_2^2$. For a Boolean-valued function on the unbiased cube, it is the probability that flipping coordinate $i$ changes the function value.

The total influence is

For the indicator $f$ of a monotone family under a p-biased product measure, the Margulis-Russo formula identifies the derivative of ${\mathbb{E}}_{{\mu }_p}f$ with the suitably normalized total influence of $f$.