Make sure you have the same filters applied in both sessions and reports.

eV-actual+actual = eV

yup, good point.

Yes.. it's an unbiased estimator.

can you elaborate why?

Sure. Say you play for n hands and the amount that you win/lose is giving by the random variable WIN. Say that the amount that you win in all-in pots is given by ALL.

Then of course we can write WIN = (WIN-ALL) + ALL. We take the EV of both sides, EV(WIN) = EV((WIN-ALL) + ALL) = EV(WIN-ALL) + EV(ALL), because EV is linear.

Then consider the expression (WIN-ALL) + EV(ALL). That's the definition of AIEV. We take the expected value of that and find EV(AIEV) = EV((WIN-ALL)+EV(ALL)) = EV(WIN-ALL) + EV(EV(ALL)) = EV(WIN-ALL) + EV(ALL). I used the fact that EV is linear and the fact that EV(EV(X)) = EV(X) (forget what that property is called...).

Then all you gotta do is notice that the right sides of the two different things i derived are the same, so we have EV(WIN) = EV(AIEV). That's the definition of an unbiased estimator.

Looks good to me, thanks!

E[X] is a constant and expectation of constants are themselves.

well, yeah, but there's a name for a function f such that f(f(x)) = f(x) for all x.

Not sure. Looks like a generalization of eigenvectors/eigenvalues, i.e Ax = x.

What are non-linear examples of f(f(x)) = f(x)?

There are tons of them. For example let f(x) = |x|. Or let f(x) = x if x is not an integer, f(x) = x+1 if x is an even integer and f(x) = x-1 if x is odd. Or let f(x) take the decimal representation of x and swap all the 2s for 7s and 7s for 2s. Etc, etc etc.

There are nice theorems about them. Wish i could remember the name . Google is not helping.

Idempotent (IIRC).

Juk

nah.. idempotent means f^n = id. Not quite the same .

Are you talking about Iterated Expectation,

E[X] = E[E[X | Y]]

where in this case the auxiliary random variable Y is X?

Neither of those have the property that f(f(x))=f(x). I think you might have been thinking of f(f(x) = x.

I would infer from the definition that on the range of f, f is the identity. In fact this is an equivalent definition.

In symbols:
f: A -> B
f | f(A) = I(A)

Clearly B is a subset of A, for the definition to make sense.

Oops.. I'm an idiot. I accidentally gave idempotent examples :/. That's prob why juk thought I was talking about that. This is what I meant to write:

1) f(x) = |x|
2) f(x) = x if x is not an even integer, f(x) = x+1 if x is an even integer
3) Let f(x) take the decimal representation of x and swap all the 2s for 7s

Some other non-trivial examples:

4) f(x) = sign(x) (i.e. f(x) = -1 if x is negative, +1 if x is positive, and 0 if x is 0)
5) f(X) = EV(X) (hey remember that one?!)
6) The linear operator represented by a matrix with 0s off the diagonal and either 1s and 0s on the diagonal. For example:
(1 0)
(0 0)

Your definition is indeed equivalent. All of the examples that I've given have really obvious ranges because it's pretty easy to come up with examples by using your equivalent definitions. However, I think that my definition is the standard one because there are many important examples (none of which I remember ) that really weird ranges but still clearly satisfy f = f^2. So, a math text that discusses this (I think it probably comes up in linear algebra and ring theory most often) might start with:

Definition 1.1: A function f is said to be word Noah can't think of if f = f^2. (Where f^2 is my ASCII way of writing f composed with f.)

Lemma 1.2: Let f : A -> A be a function such that f restricted to f(A) is the identity. Then f is word Noah can't think of.


Ugh.. sorry for the weird hijack. This is gonna drive me nuts.

Really appreciate the familiarity with math/stats ITT. Just had to say that.