Mean/Variance Questions

This "question" is really two seperate questions:

1) What is the standard deviation for a typical winning limit hold-em player at middle limits? I have heard 10, 15 and 20 BB, so would anyone like to clarify?

2) Thinking back to my statistics classes I remembered this line, "The variance/mean of a sum of independent random variables is the sum of the individual variances/means." Now my question: would this theorem apply to multi-tabling? To elaborate, say I have a u=1/hr (mean) and a var=225/hr per table, and consequently a std. dev.=15/hr. Does that imply that two-tabling I would have a u=2/hr, var=450/hr, and std. dev.=21.21/hr? Thanks all.

[BruceZ]

[ QUOTE ]
This "question" is really two seperate questions:

1) What is the standard deviation for a typical winning limit hold-em player at middle limits? I have heard 10, 15 and 20 BB, so would anyone like to clarify?

[/ QUOTE ]

It depends on whether you're talking per hour or per 100 hands. When playing live, the benchmark has been 10 times your win rate per hour, so a win rate of 1 bb/hr would correspond to a SD of 10 bb. For 100 hands, this would correspond to a win rate of about 3 bb/100, and a SD of 17 bb, hence you might see SD values of 15-20 bb for 100 hands. It may be significantly higher if you are playing shorthanded, or in very aggressive games.


[ QUOTE ]
2) Thinking back to my statistics classes I remembered this line, "The variance/mean of a sum of independent random variables is the sum of the individual variances/means." Now my question: would this theorem apply to multi-tabling? To elaborate, say I have a u=1/hr (mean) and a var=225/hr per table, and consequently a std. dev.=15/hr. Does that imply that two-tabling I would have a u=2/hr, var=450/hr, and std. dev.=21.21/hr? Thanks all.

[/ QUOTE ]

Correct. Note that by playing 2 tables at half the limit, you can reduce your standard deviation per hour without reducing your win rate, and hence greatly reduce your risk of ruin for a given bankroll, or cut your bankroll requirement in half for a given risk of ruin. This assumes that you can make the same number of bb/hr at each table at the lower limit as you did at the higher limit.

Thanks Bruce. By the way, I remember a while back you provided the "risk of ruin" equation as RoR=e^(-2uB/var). How is that equation derived? It looks somewhat related to the pdf for the normal distribution function, but I'm not sure how to get from one to the other (derivative, integral, other).

[BruceZ]

[ QUOTE ]
Thanks Bruce. By the way, I remember a while back you provided the "risk of ruin" equation as RoR=e^(-2uB/var). How is that equation derived? It looks somewhat related to the pdf for the normal distribution function, but I'm not sure how to get from one to the other (derivative, integral, other).

[/ QUOTE ]

Derivation of risk of ruin formula