normal distribution

[irchans]

jason1990 writes

[ QUOTE ]
Does anyone know, roughly, how many poker hands a person needs to play before their winnings become approximately normally distributed? For example, suppose someone has a winrate of 2BB/100 and a standard deviation of 15BB/100. They might think that if they play 100 hands, then they have a 95% chance of winning somewhere between -28BB and 32BB (a spread of 2 SD's). But this is only true if the result of playing 100 hands is (roughly) normally distributed. Is it?


[/ QUOTE ]

This brings up some interesting questions about the sum of identically distributed random variables.

Given identically distributed random variables x_i with 0 mean and standard deviation s. Let the random variable y_N be defined to be

y_N = (x_1 + x_2 + ... + x_N)/Sqrt(N).

By the central limit theorem, P( y_N > 2 s) converges to (1-Erf(Sqrt(2)))/2 = 0.02275 as N becomes large, but the theorem does not tell us how quickly it converges.

Question 1:
Is there a bound on P( y_N > 2 s), the probability that y_N will be more than 2 standard deviation above the mean?

Question 2: If we know that P( |x_i| > a) = 0 for some real number a, is there a sharper bound on P( y_N > 2 s) using a?