For those who don't think 30+ BI swings are a reality...

[bilbo-san]

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I see that, but I don't see how that has anything to do with how fat the tails of the distribution are.

Please correct me if I'm wrong but if the tail is fatter the chance of a sample far from the mean is larger, right?

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I think a winning player has a fat tail to the right, and a thin tail to the left of the mean.

The more winning a player is, the more extreme this is.

Does that make more sense? Does that illustrate how negative swings become pretty unlikely the higher the winrate?

[Pomtidom]

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Does that make more sense? Does that illustrate how negative swings become pretty unlikely the higher the winrate?

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I completely understand that already. [img]/images/graemlins/smile.gif[/img]

The thing I'm trying to understand is, the OP used a normal distribution to render some 100k hands examples. Now Pokey stated that the 'poker distribution' would have fatter tails, now a distribution with fatter tails should cause bigger swings, no?

I don't know if this is a good reference but the tails of the Cauchy distribution are so 'fat' it's mean is even undefined.

I'm no math expert by any means, so please correct me if I'm way off.

[Bottled Rockets]

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This thread is great - Pokey you do a great job of explaining these stats concepts.

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QFT

Very interesting thread!

[anacrime]

Best Thread Ever

[DrMagic]

Great thread guys. Excellent posts Pokey and BadMongo. I suck at stats so this thread is a real eye-opener.

[bruin]

this makes me feel a lot better about the 25 BI I dropped 2/4 nl, lol.

[good2cu]

i think there is something very wrong with these graphs.

[ffredd]

BadMongo, can you explain the formula you're using in the simulation?

=SQRT(-2*LN(RAND()))*SIN(2*PI()*RAND())

I'm good at other kinds of math, but I suck at statistics. So don't hesitate to get technical, but also don't assume that I already know every theorem in statistics.

[Kristian]

BM, Pokey, great thread/discussion. Sorry for joining so late.

You both seem to agree that we need to specify the distribution of a single hand in order to model the path of 100k hands. You are obviously correct on a theoretical level, but I was wondering if the OP model isn't in fact very close to realistic (using proper SD numbers of course).

What we need to adress is how close to normally distributed a 100 hand sample actually is. Have you done this?

If it's pretty close, the improvement achieved by modelling the precise hand distribution will be very small, and we can safely use the existing model.

If it's not close, we could try using 200 or 500 hand samples to achieve better fits. The Central Limit Theorem tells us that it is only a question of using a large enough sample size. Of course using very large chunks of hands to get a proper fit to a normal distribution will cause us to lose detail, but perhaps this is not necessary?

[Kristian]

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No, that's my whole point: the distribution is NOT normal. I'm agreeing with you that some descriptive statistics of a non-normal distribution (like its mean) are themselves normally distributed (CLT gives us that), but the distribution itself is not. For example, if you take 1000 samples from a uniform distribution, the sample mean will converge towards the population mean and the sample mean's value will be normally distributed. That does NOT mean that the underlying distribution itself is normally distributed -- it's still uniformly distributed. What you're doing in your spreadsheet is like saying "the mean of a sample population drawn from uniformly distributed population will be normally distributed; therefore, if I know the mean and standard deviation of my sample I can simulate draws from that distribution by randomly pulling from a normal distribution with the same mean and standard deviation." Your mean and standard deviation may be complete accurate, but your simulator won't work because it's using the wrong distribution.


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What seems to be forgotten in this discussion is that according to the Central Limit Theorem, the SUM of the 1000 samples will be approximately normally distributed, no matter the distribution of each individual sample. If the approximation is bad, we simply need to use more samples.

Correspondingly, if our 100 hand samples are not close to normally distributed, we should increase the number of hands in each chunk, and the accuracy of the model will increase.

This seems to be an easier approach than trying to come up with a precise hand distribution, and then simulating 100k of those.