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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.

A site $v$ is pivotal for an increasing event $E_N$ in a configuration $\omega$ when changing only the state of $v$ changes whether $E_N$ occurs. Away from the boundary, pivotality for a left-to-right crossing has the geometric four-arm description: from neighbors of $v$ there are two disjoint open arms reaching the left and right sides and two disjoint closed arms reaching the top and bottom sides, in alternating cyclic order. The boundary version truncates the corresponding arms at the sides already adjacent to $v$.

The Margulis–Russo formula gives