Standard Deviation Question
 

Let's say you expect to win a certain bet 60% of the time. You would lose 40% of the time. To find out your SD for 10 bets you would take the square root of (10 x .6 x .4)= 1.549

You would then multiply this times 3 for for 3 SDs = 4.67

Now to find your range of winners you would subtract 4.67 form 6 and add 4.67 to 6 getting 1.33 and 10.67.

Within 3 SDs you could expect to win this bet anywhere from 1.33 to 10.67 times out of ten.

My question is how can you get a number over 10? I'm taking this straight from a book. It works out OK for larger sample sizes, but when I tried it for smaller ones this is what happens. I guess I don't understand what is happening here. Did I do the calculations correctly?

Thanks for any explanation.

Re: Standard Deviation Question
 

That's because the approach you use assumes the game (all 10 trials at once) has a normal distribution (with an ev of 6 and an sd of 4.67) of winners where in fact it's a binomial distribution. Now the binomial distribution converges to a normal distribution for a large number of trials, but 10 just isn't large enough.
If this were game normal distributed it would be possible (although unlikely) to get values bigger than 10 or smaller than 0.

Re: Standard Deviation Question
 

http://www.stat.wvu.edu/SRS/Modules/...malapprox.html

Don't think the binomial converges to the normal.
The normal is just a good approximation of the binomial.

Re: Standard Deviation Question
 

But the distribution of sums of enough bernoulli-distributed random variables converges to a normal distribution due to the central limit theroem. And the sum of bernoulli-distributed random variables has a binomial distribution.

Re: Standard Deviation Question
 

[ QUOTE ]
But the distribution of sums of enough bernoulli-distributed random variables converges to a normal distribution due to the central limit theroem. And the sum of bernoulli-distributed random variables has a binomial distribution.

[/ QUOTE ]

wow...is this English, ...i feel so dumb...but truly i am impressed


(although it does sound sarcastic, and i don't mean it to)

Re: Standard Deviation Question
 

[ QUOTE ]
http://www.stat.wvu.edu/SRS/Modules/...malapprox.html

Don't think the binomial converges to the normal.
The normal is just a good approximation of the binomial.

[/ QUOTE ]
eh, doesn't everything converge to the normal?

Re: Standard Deviation Question
 

[ QUOTE ]
wow...is this English, ...i feel so dumb...but truly i am impressed

[/ QUOTE ]

i has no clue, i is german [img]/images/graemlins/smile.gif[/img]

Re: Standard Deviation Question
 

[ QUOTE ]
[ QUOTE ]
http://www.stat.wvu.edu/SRS/Modules/...malapprox.html

Don't think the binomial converges to the normal.
The normal is just a good approximation of the binomial.

[/ QUOTE ]
eh, doesn't everything converge to the normal?

[/ QUOTE ]

The Poisson is used for expectation of wins in tourneys if your name isn't Stu Unger.
Maybe for very large n it converges. Only your skill differential from the field doesn't remain constant over time.

Re: Standard Deviation Question
 

[ QUOTE ]
[ QUOTE ]
http://www.stat.wvu.edu/SRS/Modules/...malapprox.html

Don't think the binomial converges to the normal.
The normal is just a good approximation of the binomial.

[/ QUOTE ]
eh, doesn't everything converge to the normal?

[/ QUOTE ]

No.

Re: Standard Deviation Question
 

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
http://www.stat.wvu.edu/SRS/Modules/...malapprox.html

Don't think the binomial converges to the normal.
The normal is just a good approximation of the binomial.

[/ QUOTE ]
eh, doesn't everything converge to the normal?

[/ QUOTE ]

No.

[/ QUOTE ]
okay, what doesn't?

Re: Standard Deviation Question
 

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
http://www.stat.wvu.edu/SRS/Modules/...malapprox.html

Don't think the binomial converges to the normal.
The normal is just a good approximation of the binomial.

[/ QUOTE ]
eh, doesn't everything converge to the normal?

[/ QUOTE ]

No.

[/ QUOTE ]
okay, what doesn't?

[/ QUOTE ]

Anything with non-finite variance, like for example poker and the majority of things that you encounter in your daily life. IMO, anyone who talks about normal distributions or the Central Limit THeorem on a poker website should be banned.

A much shorter list are the things that do converge to the normal.

Re: Standard Deviation Question
 

[ QUOTE ]
Anything with non-finite variance

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What do you mean by this? flipping a coin, winning/losing a dollar per flip, forever has non-finite variance, no?

or do you mean non-finite variance per event or whatever? how does poker not have finite variance per hand if this is what you mean?

Re: Standard Deviation Question
 

Coin flipping has finite variance.

Poker's variance has infinite possibilities.

Re: Standard Deviation Question
 

[ QUOTE ]
Coin flipping has finite variance.

Poker's variance has infinite possibilities.

[/ QUOTE ]
I'm still not getting this

coin flipping has infinite possibilities too. The variance I believe is roughly proportional to the square root of the number of flips, hence it tends to infinity if the number of flips tends to infinity

Re: Standard Deviation Question
 

poker = continuous
coin flipping = discrete

Re: Standard Deviation Question
 

[ QUOTE ]
poker = continuous
coin flipping = discrete

[/ QUOTE ]
what does this imply?

please spell it all out for me

Re: Standard Deviation Question
 

Poker results should follow normal distribution (a normal distribution is continuous). Coin flipping results follow a binomial distribution (a binomial distribution is discrete).

A distribution is continuous if there are an infinite number of possible outcomes over a finite region. It might make more sense if you think about radios. The FM radio in your car is discrete (you can only tune it to certain frequencies on the interval (88.1, 107.9). The crappy alarm clock radio you had as a kid, that was a pain in the ass to tune was continuous. You could tune it to any frequency you wanted on the interval (88.1, 107.9), even frequencies like 99.7966445.

The point is discrete distibutions always stay discrete. After a while they look a lot like continuous ones and can be approximated very well by them, but they're still not continuous.

Re: Standard Deviation Question
 

oh right, I thought it was something more profound. What if I change my question to:

"eh, doesn't everything converge to the normal or binomial?"

also, is Troll_Inc making any sense when he says:

[ QUOTE ]
Anything with non-finite variance, like for example poker and the majority of things that you encounter in your daily life. IMO, anyone who talks about normal distributions or the Central Limit THeorem on a poker website should be banned.

A much shorter list are the things that do converge to the normal.

[/ QUOTE ]

Re: Standard Deviation Question
 

Van nostrin is talking about the wrong moment and is confusing you.

Re: Standard Deviation Question
 

[ QUOTE ]
Coin flipping has finite variance.

Poker's variance has infinite possibilities.

[/ QUOTE ]

Having infinite possibilities doesn't imply infinite variance. And the distribution of the outcome of a single hand in NLH has finite variance unless you assume two players have infinite bankrolls.
Thus the outcome of n hands will converge to a normal distribution with large n. But this convergence is very much slower than most people suspect it to be. I.e. BB/100 is by no means normal, BB/1000 isn't, BB/1m probably comes close.

Re: Standard Deviation Question
 

[ QUOTE ]
[ QUOTE ]
Coin flipping has finite variance.

Poker's variance has infinite possibilities.

[/ QUOTE ]

Having infinite possibilities doesn't imply infinite variance. And the distribution of the outcome of a single hand in NLH has finite variance unless you assume two players have infinite bankrolls.
Thus the outcome of n hands will converge to a normal distribution with large n. But this convergence is very much slower than most people suspect it to be. I.e. BB/100 is by no means normal, BB/1000 isn't, BB/1m probably comes close.

[/ QUOTE ]

Same problem here.....starts talking about variance then stumbles into the mean (average).

Re: Standard Deviation Question
 

[ QUOTE ]
starts talking about variance then stumbles into the mean (average).

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wtf? care to elaborate?

Re: Standard Deviation Question
 

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A much shorter list are the things that do converge to the normal.


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Btw this statement is wrong.

Re: Standard Deviation Question
 

[ QUOTE ]
[ QUOTE ]
A much shorter list are the things that do converge to the normal.


[/ QUOTE ]

Btw this statement is wrong.

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I'm so right it's laughable.

I'll tell you what. For evey non-normal daily distribution I name you have to name a normal one. When you give up, I'll name an extra 30. LOL.

Non normal 1. The weight of items in my kitchen. Totally non-normal. I have tons of small items and very few large.

Your turn.

Re: Standard Deviation Question
 

You obviously don't understand what the phrase 'converges to a normal distribution' means. The weights of items from a given set don't converge or whatsoever. And nobody ever claimed that every distribution is normal.

Re: Standard Deviation Question
 

[ QUOTE ]
You obviously don't understand what the phrase 'converges to a normal distribution' means. The weights of items from a given set don't converge or whatsoever. And nobody ever claimed that every distribution is normal.

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You don't understand the difference between mean and variance as evidenced by your post above. You start talking about variance and then even in the same paragraph all the sudden start talking about means.

I also didn't say that everything is non-normal. There are just way more non-normal distributions than normal. I already proved my argument apparently as you can't name a single normal distribution and already want to change your argument.

Re: Standard Deviation Question
 

Also you're obviously not familiar with the phrase 'by no means'.

And I don't know which argument you want to 'prove' here - that most distributions aren't normal? That's a pretty obvious fact. That most distributions don't converge to a normal distribution?

The CLT states sth. like: The normalized sum of independent identically distributed random variables converges to a normally distributed random variable.

Re: Standard Deviation Question
 

[ QUOTE ]
Also you're obviously not familiar with the phrase 'by no means'.

And I don't know which argument you want to 'prove' here - that most distributions aren't normal? That's a pretty obvious fact. That most distributions don't converge to a normal distribution?

The CLT states sth. like: The normalized sum of independent identically distributed random variables converges to a normally distributed random variable.

[/ QUOTE ]

The CLT doesn't rule my kitchen!

Re: Standard Deviation Question
 

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

Re: Standard Deviation Question
 

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]

eh, doesn't everything converge to the normal?

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No.

[/ QUOTE ]
okay, what doesn't?

[/ QUOTE ]
Here is an example. An object is located at the origin of the number line. There are n other objects with the same mass distributed uniformly on the interval [-n,n]. Let us estimate the net gravitational force on the object at the origin, assuming n is large.

Let X_j denote the location of the j-th object. Then the net force on the center object is proportional to

F_n = \sum_1^n sgn(X_j)/|X_j|^2.

For simplicity, let us take the constant of proportionality to be 1. We can write X_j = nY_j, where Y_j are iid uniform [-1,1]. Then F_n = n^{-2}\sum_1^n Z_j, where

Z_j = \sgn(Y_j)/|Y_j|^2.

Note that E|Z_j| = infinity, so these variables do not even have a first moment, let alone a second moment. Hence, we cannot apply the central limit theorem. As it turns out, as n goes to infinity, the distribution of F_n converges to a distribution with characteristic function e^{-c|t|^{1/2}}, for some constant c. For comparison, the characteristic function of the standard normal is e^{-0.5|t|^2}.

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

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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.