Consider a system with $n$ units where each unit is either working or failing. $x_j=1$ represents the condition that $j$-th unit is working. Suppose each unit is working with independent probability $p_j$, then show that the Boolean function $f(x_1, x_2, \dots x_n)$ is true with probability $F(p_1, p_2, \dots p_n)$ where $F$ is a polynomial in the variables in $p_1, p_2, \dots, p_n$.
I haven't understood the question properly. I understand what the Boolean variable $x_j$ means but what does the Boolean function $f$ mean and how do we prove anything about its probability ?
The solution uses the fact that $F$ is a multi linear integral representation of $f$, and that proves it. Now, I've read that a multilinear representation of a Boolean function is a sum of zero or more of its $2^n$ possible terms mod $2$. Now, an integral multilinear representation is this same multilinear representation such that each term is weighted with an integral coefficient so that $f(x_1,\dots x_n)$ has the correct value at all $2^n$ possible vectors without taking the reminder mod 2.But I don't understand how it relates. Please help.
My main questions are : 1. What is this Boolean function $f$ ? There can be any number of Boolean functions. How to prove something in general for all ?
- The concept of multilinear integral representation isn't clear. The author just said that a polynomial in its probabilities will be a multilinear integral representation of the Boolean function and that's that.