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Continuing from my previous question on Brunner et al.'s paper; so given a standard Bell experimental setup:

where independent inputs $x,y \in \{0, 1\}$ decide the measurement performed by Alice & Bob on quantum state $S$ with outcomes $a,b \in \{-1, 1\}$, $a$ and $b$ are correlated (not independent) if:

(1) $P(ab|xy) \ne P(a|xy)P(b|xy) \ne P(a|x)P(b|y)$

Of course, there are perfectly innocent non-quantum reasons why $a$ and $b$ could be correlated; call these reasons confounding variables, some artifact of when Alice & Bob's systems interacted in the past. The set of all confounding variables we call $\lambda$. If we take into account all variables in $\lambda$, a local theory claims that $a$ and $b$ will become independent and thus $P(ab|xy)$ will factorize:

(2) $P(ab|xy,\lambda) = P(a|x,\lambda)P(b|y,\lambda)$

This equation expresses outcomes depending only on their local measurement and past variables $\lambda$, and explicitly not the remote measurement.

**Question one**: what is the mathematical meaning of the comma in equation (2)?

**Question two**: what is an example of a variable in $\lambda$?

The paper then says the following:

The variable $\lambda$ will not necessarily be constant for all runs of the experiment, even if the procedure which prepares the particles to be measured is held fixed, because $\lambda$ may involve physical quantities that are not fully controllable. The different values of $\lambda$ across the runs should thus be characterized by a probability distribution $q(\lambda)$.

**Question three**: why was it a set of variables before but is now only a single variable?

**Question four**: what is an example of a probability distribution for $q(\lambda)$ here?

We then have the fundamental definition of locality for Bell experiments:

(3) $P(ab|xy) = \int_{Λ} q(\lambda)P(a|x, \lambda)P(b|y, \lambda) d\lambda$ where $q(\lambda|x,y) = q(\lambda)$

**Question five**: What does the Λ character mean under the integral sign?

**General question**: so we have a continuous probability distribution $q(\lambda)$ over which we're integrating. Why are we multiplying the RHS of equation (2) by $q(\lambda)$ in the integrand? That would seem to make equation (3) different than equation (2). What's an example of this integral with concrete values?