I'm reading a physics paper (John Bell's 1964 paper on the EPR paradox if anyone is physics-curious) and I'm having an issue following his derivation. It's the probability distribution stuff -not totally complicated, I'm sure- that's giving me trouble...
$\overrightarrow{\sigma}$ is a hypothetical measurement of a spin-value of a particle (say, an electron). It's inner product with a given unit vector (say, $\overrightarrow{a}$) would only yield values of $\pm1$. $\,\lambda$ is in this context a hypothetical 'hidden variable', it could be a single value or a function (he claims that this is irrelevant for the argument). The result of measuring $\overrightarrow{\sigma_1}\cdot\overrightarrow{a}$ is then determined by $\overrightarrow{a}$ and $\overrightarrow{\lambda}$ and the result of $\overrightarrow{\sigma_2}\cdot\overrightarrow{b}$ in the same instance is determined by $\overrightarrow{b}$ and $\overrightarrow{\lambda}$, and
$$A(\overrightarrow{a},\lambda) = \pm1, \,\,B(\overrightarrow{b},\lambda) = \pm1$$
If $\rho(\lambda)$ is the probability distribution of $\lambda$ the expectation value of the product of the two components $\overrightarrow{\sigma_1} \cdot$ $\overrightarrow{a}$ and $\overrightarrow{\sigma_2} \cdot \overrightarrow{b}$ is
$$ P(\overrightarrow{a},\overrightarrow{b}) = \int d\lambda\rho(\lambda)A(\overrightarrow{a},\lambda)B(\overrightarrow{b},\lambda)$$
Let the "hidden variable" be (for example) a unit vector $\overrightarrow{\lambda}$ with uniform probability distribution over the hemisphere $\overrightarrow{\lambda}\cdot\overrightarrow{p} > 0$. Specify that the result of measurement of a component $\overrightarrow{\sigma}\cdot\overrightarrow{a} $ is
$$sign \overrightarrow{\lambda}\cdot\overrightarrow{a'},$$
where $\overrightarrow{a'}$ is a unit vector depending on $\overrightarrow{a}$ and $\overrightarrow{p}$ in a way to be specified, and the sign function is +1 or -1 according to the sign of the argument...Averaging over $\overrightarrow{\lambda}$ the expectation value is
$$<\overrightarrow{\sigma}\cdot\overrightarrow{a}>\, =\, 1 - \frac{2\theta'}{\pi}\,\,\,\,\,\,[1]$$
It's this last part [1] that I'm having trouble with. I don't see how he pulled that as the expectation value. I know this is more of a physics problem. But I figured the math community could give me some good insight. I know this is lengthy and very 'physics', but thanks for the help in advance.
Here's the paper if anyone wants to see it. The derivation/argument begins on the bottom of page 1 to the bottom of page 2.