 Margulis–Russo formula

For an increasing Boolean function $f:\{0,1{\}}^n\to \{0,1\}$ and independent Bernoulli distribution input bits of probability $p$, define $I_i^{(p)}(f)$ as the probability that coordinate $i$ is pivotal for an increasing event. Then, for $0<p<1$,To prove it, first give each coordinate its own parameter $p_i$. Conditioning on all other coordinates makes the expectation affine in $p_i$, with partial derivative equal to the expected difference between the values with bit $i$ fixed to one and zero. Monotonicity makes that difference the pivotal indicator. The chain rule along $p_1=\cdots =p_n=p$ yields the formula.^[ref]

For example, an AND of $n$ input bits has expectation $p^n$ and each pivotal probability is $p^{n-1}$. In the sign-coordinate p-biased product measure, $p$ instead denotes the mass at minus one: identify the increasing bit as $(1-x_i)/2$. Keeping an increasing function of the ordered signs themselves reverses the derivative's sign. Biased squared-derivative definitions of total influence also require their stated normalization.