The writings do not specify the marginal distributions of theta_0 and e_0, let along the joint distribution. The various people writing programs have taken uniform distributions, and independence. Sometimes discrete uniform instead of continuous uniform. Sometimes e_0 is taken from the circle instead of from the sphere.
Minkwe: Pick a large integer K and let delta = (pi/2)/K. Picking an angle t uniformly at random from the finite set {0, delta, 2 delta, ... , pi/2} and calculating 1/2 sin(t)^2 is not much different from picking an angle t uniformly at random from the interval [0, pi/2] and calculating 1/2 sin(t)^2.
Are you saying that Nature uses the first method? Where is it specified in Joy's writing that the angle theta_0 comes from a discrete distribution? What is the value of K?
When the sample size is large enough and the increments between Alice and Bob's angles are small enough, and when you zoom in and look at the graph at, say, 30 degrees, you'll see a small but systematic deviation.
This is the value at 30 degrees, to three decimals: -0.849
(and without any articial discretizations; computed with John Reed's Mathematica code)
This is negative cosine of 30 degrees, to three decimals: -0.866
Fred, Joy: please show us pictures, not of the whole curves (theoretical and experimental), but just the bits between, say, 29 and 31 degrees. On the vertical scale: between -0.9 and -0.8
If you take the measurement angles all to be whole numbers of degrees then all differences between measurement angles are whole numbers too. You can plot just three points on each of the two curves for us.
Here is an example from http://rpubs.com/gill1109/EPRB2 (minkwe's model in R, upside down). Correlations are computed for differences between Alice and Bob's angles equal to 0 degrees, 7.5 degrees, 15 degrees .... Sample size is 1 million, at each measurement angle pair.. There is *no* unnecessary disretization: this is as accurate as it can get.
The title of the plot should read "negative of two correlation functions".
