mean]

2026-08-27

The mean of a function $𝑓 : {\pm 1}^{𝑛} \to ℝ$ is $𝐄 [𝑓]$ ; i.e.,

(\frac{1}{2^{𝑛}}) \sum_{𝑥 \in {\pm 1}^{𝑛}} 𝑓 (𝑥),

the average value of $𝑓$ .

In particular, when we have a Boolean-valued function $𝑓 : {\pm 1}^{𝑛} \to {\pm 1}$ , then

𝐄 [𝑓] = 𝐏𝐫 [𝑓 = 1] - 𝐏𝐫 [𝑓 = - 1] .

If $𝑓$ has a mean of $0$ , we call $𝑓$ unbiased or balanced.

Note. [#]

For $𝑓 : {\pm 1}^{𝑛} \to ℝ$ ,

𝐄 [𝑓] = \hat{𝑓} (\emptyset) .

This follows from the fact that $𝐄 [𝑓] = ⟨ 𝑓, 1 ⟩ = \hat{𝑓} (\emptyset)$ .