A theorem in probability theory

[gill1109]

Fred, that was not the smartest way to solve that equation
I recommend you start by taking logs
log_e(0.1 / 8) = - N (eta/16)^2
N = log_e(800) x 160^2

About 172 thousand

You should have done Solve, not Reduce

[gill1109]

Notice: my theorem states a bound on a probability. It does not state that that bound is the best bound. I can give you much sharper but rather more complicated bounds.

I can also give you approximations. Good ones. With much smaller N, the chance that the CHSH-like quantity I defined in the theorem would be bigger than 2.1 is already very small indeed.

You could easily do simulations to verify my claims. That is also a good way to check that you understand the statement of the theorem.

[gill1109]

Enjoy: https://rpubs.com/gill1109/theorem

[FrediFizzx]

Fred, that was not the smartest way to solve that equation
I recommend you start by taking logs
log_e(0.1 / 8) = - N (eta/16)^2
N = log_e(800) x 160^2

About 172 thousand

You should have done Solve, not Reduce.

Solve doesn't find all the solutions. Not good.
.

[gill1109]

Fred, that was not the smartest way to solve that equation
I recommend you start by taking logs
log_e(0.1 / 8) = - N (eta/16)^2
N = log_e(800) x 160^2

About 172 thousand

You should have done Solve, not Reduce.

Solve doesn't find all the solutions. Not good.
.

A little thought will show that there is exactly one real solution to that equation. Solve found it for you, up to whatever numerical accuracy you asked for. The smallest integer at least as large as the unique real solution is what you really want. Well done.

The answer is N = 171127 or larger.

Code: Select all

> 8 * exp(-171126 / 160^2)
[1] 0.01000002
> 8 * exp(-171127 / 160^2)
[1] 0.009999633

[gill1109]

Mathematica's warning message is misleading. We are working over the real numbers, when we try to solve that equation. The inverse of the function "exp" is "log". These are strictly monotone, continuous, real functions. The equation is easily seen to have exactly one solution. You can easily write down the formula for it. Mathematica probably can too, but it apparently needs to be given some help.

Actually, we want to find the smallest positive integer which satisfies an *inequality*, not an "equality*.

Mathematica programming challenge: see if you can tell Mathematica to do what we actually want it to do!

[FrediFizzx]

@gill1109 None of this really matters any more. New CHSH 10 run average,

CHSH = 2.81796!
.

[gill1109]

More maths: https://ieeexplore.ieee.org/document/9622238

[FrediFizzx]

More maths: https://ieeexplore.ieee.org/document/9622238

You really should stop writing nonsense papers and posting nonsense to the forum.
.