The Mathematics of Poker

[thylacine]

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thylacine said:[ QUOTE ]
What on earth is the quantity that equals 1.61 on page 34? [img]/images/graemlins/mad.gif[/img]

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Oct0puz said:[ QUOTE ]
"As we calculated above, the standard deviation of a sample of 16,900 hands is 1.61 units/100hands" Thats what it is. Or is there something in the quote that is unclear?

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thylacine said:[ QUOTE ]
Yes, we are looking at the same book. The quote is about as clear as pea-soup flavored porridge in a thick fog.

Let me ask again. What on earth is the quantity that equals 1.61 on page 34? What is the precise mathematical meaning of each of the quantities in this calculation?

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thylacine says: Even more to the point: What exactly is being done to a random variable with standard deviation 2.1 to produce another random variable with standard deviation 1.61?

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The SD of 1.61 is a renormalization of the standard deviation of a 16,900 hand sample into BB/100h terms. It's not the standard deviation of a 100 hand sample, of course! It's actually not a random variable, either.

Maybe this would be clearer if we said the following instead -- we won 1.15 BB/100 hands in 16,900 hands, which is 194.35 BB. Our standard deviation for a sample of this size is 273 BB, and change all the confidence interval stuff to read "194BB +/ 273BB" etc.

It's just that it seemed that people are more comfortable with BB/100 as a metric than large raw numbers of big bets over 16,900 hands.

Jerrod

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Okay let me try to reverse engineer what you are saying. Tell me if this is right.

The question is what is the significance of the quantity

S(\sigma,N,K) := K \sigma / \sqrt{N} ---(EQ1)

(Note \sqrt{N} in denominator.) In particular you had \sigma=2.1 N=16900 K=100. This quantity is also given by

\sigma \sqrt{N} / H where H=N/K ---(EQ2)

(Note \sqrt{N} in numerator.) In particular you had H=169.

When K=N, H=1 then the quantity is

\sigma \sqrt{N} ---(EQ3)

which is what is in your equation (2.4) so no problem there. (Note \sqrt{N} in numerator.) But if K=1, H=N then the quantity is

\sigma / \sqrt{N} ---(EQ4)

(Note \sqrt{N} in denominator.) The question is, what is the significance of this quantity and why can we scale by multiplying by K to give a per-K-hand version, which is clearly quite different to your equation (2.4).

So I guess it is this. With K=1 or 100 or 16900 or anything you like, from the statstics gathered (and using "classical method") the "best guess" for the "true underlying mean" is 0.0115K per K hands (which scales as you would expect for the mean of the sum of K samples) and that you could represent your uncertainty about what the "true underlying mean" is as a "standard deviation of the best guess of the "true underlying mean" per K hands" is the abovementioned quantity

K \sigma / \sqrt{N}

(which equals)

\sigma \sqrt{N} / H

The scaling factor K is what you would expect for a mean (or a variance) but not a standard deviation. But since it is actually a "standard deviation of ... mean" it makes sense.

Does this make sense?

Do you understand why the factor K bothered me?