Re: Standard Deviation Question
 

Shouldn't E[Z_j] = 0?

The variance is still \infty, so yeah, we cannot apply the CLT here.

Re: Standard Deviation Question
 

For any random variable X, E[X] is only defined when E[|X|] is finite. So in this case, E[Z_j] is undefined. It is similar to to the fact that, in elementary calculus, the (improper) integral from -1 to 1 of 1/x is not 0; it is undefined.

Re: Standard Deviation Question
 

True, my bad. I like the example.

Re: Standard Deviation Question
 

[ QUOTE ]
i'm glad it's not just me who has no idea what troll is talking about. possibly trolling? but who would troll a stats forum? :/

btw troll the CLT does rule your kitchen

[/ QUOTE ]

The CLT rules neither my kitchen nor poker results.

It also doesn't rule wind speed, the weather, or the probability distribution of an injurious outcome the next time you get in your car.

Additionally it doesn't rule the internet which you and I are using right now.

It does rule heights of people living in this country and intelligence, which is kind of ironic given the smart people on this forum suckered into putting their faith in it.

Re: Standard Deviation Question
 

Troll_inc: you list things which are not themselves normally distributed. This is not the same thing as saying the central limit theorem doesn't apply. The CLT says that if you add up random variables which are identically distributed--whatever that distribution looks like--the distribution of the sum will approach the normal. The long run record record of a poker player is the result of adding contributions from very many hands, and each hand is independent and governed by the same distribution. You are right in saying that the distribution of the individual events must have finite variance, but each poker hand does belong to a distribution with a finite variance. Clearly, if you have an expectation E of winning a given hand, the fraction of hands that you do end up winning if you were to replay the hand over and over would tend to E.

Re: Standard Deviation Question
 

[ QUOTE ]
each poker hand does belong to a distribution with a finite variance.

[/ QUOTE ]

It does? Please cite the evidence that supports your claim.