Quick Links You last visited December 7, 2016, 1:12 pm All times shown are Eastern Time (GMT5:00)  The law of large numbers.rainbow lake Canada Member #25177 November 2, 2005 10764 Posts Offline  Posted: August 14, 2006, 7:50 pm  IP Logged  
The weak lawThe weak law of large numbers states that if X_{1}, X_{2}, X_{3}, ... is an infinite sequence of random variables, where all the random variables have the same expected value μ and variance σ^{2}; and are uncorrelated (i.e., the correlation between any two of them is zero), then the sample average converges in probability to μ. Somewhat less tersely: For any positive number ε, no matter how small, we have ProofChebyshev's inequality is used to prove this result. Finite variance (for all i) and no correlation yield that The common mean μ of the sequence is the mean of the sample average: Using Chebyshev's inequality on results in This may be used to obtain the following: As n approaches infinity, the expression approaches 1. Proof ends here The result holds also for the 'infinite variance' case, provided the X_{i} are mutually independent and their (finite) mean μ exists. A consequence of the weak law of large numbers is the asymptotic equipartition property. The strong lawThe strong law of large numbers states that if X_{1}, X_{2}, X_{3}, ... is an infinite sequence of random variables that are pairwise independent and identically distributed with E(X_{i}) < ∞ (and where the common expected value is μ), then i.e., the sample average converges almost surely to μ. If we replace the finite expectation condition with a finite second moment condition, E(X_{i}^{2}) < ∞ (which is the same as assuming that X_{i} has variance), then we obtain both almost sure convergence and convergence in mean square. In either case, these conditions also imply the consequent weak law of large numbers, since almost sure convergence implies convergence in probability (as, indeed, does convergence in mean square). This law justifies the intuitive interpretation of the expected value of a random variable as the "longterm average when sampling repeatedly". A weaker law and proofProofs of the above weak and strong laws of large numbers are rather involved. The consequent of the slightly weaker form below is implied by the weak law above (since convergence in distribution is implied by convergence in probability), but has a simpler proof. Theorem. Let X_{1}, X_{2}, X_{3}, ... be a sequence of random variables, independent and identically distributed with common mean μ < ∞, and define the partial sum S_{n} := X_{1} + X_{2} + ... +X_{n}. Then, S_{n} / n converges in distribution to μ. Proof. (See [1], p. 174) By Taylor's theorem for complex functions, the characteristic function of any random variable, X, with finite mean μ, can be written as Then, since the characteristic function of the sum of independent random variables is the product of their characteristic functions, the characteristic function of S_{n} / n is The limit e^{itμ} is the characteristic function of the constant random variable μ, and hence by the Lévy continuity theorem, S_{n} / n converges in distribution to μ. Note that the proof of the central limit theorem, which tells us more about the convergence of the average to μ (when the variance σ ^{2} is finite), follows a very similar approach. References Grimmett, G. R. and Stirzaker, D. R. (1992). Probability and Random Processes, 2nd Edition. Clarendon Press, Oxford. ISBN 0198536658.
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rainbow lake Canada Member #25177 November 2, 2005 10764 Posts Offline  Posted: August 14, 2006, 7:51 pm  IP Logged  
Which brings me to my next point. To be continued.   
United States Member #39406 May 15, 2006 287 Posts Offline  Posted: August 14, 2006, 8:06 pm  IP Logged  
Can you illustrate this at a little more elementary level. I flunked math. No need in me trying to learn that stuff now is it.   
rainbow lake Canada Member #25177 November 2, 2005 10764 Posts Offline  Posted: August 15, 2006, 8:30 am  IP Logged  
Here is a series of numbers. These number have paid out over 200 million dollars in the last 3 months, What is interesting about these are, read up and down instead of across one way, take first 3 digits of each series of numbers can you predict what the next 3 numbers will be, only one rule 0 can not be a first digit, 3102123334443131528293347252131925323040626273341422413162142464983715384347322692043444681321222439413172133414238626303844451611142933424472212229343571119364146493835192034473122026293745174173439414626111420253136481011363846482618233234414649111214273448882021273648111114344042436915183435456115181929375151822314349269192024364810210111642432131736444548862830313448471172429344627   
rainbow lake Canada Member #25177 November 2, 2005 10764 Posts Offline  Posted: August 15, 2006, 8:31 am  IP Logged  
3102123334443131528293347252131925323040626273341422413162142464983715384347322692043444681321222439413172133414238626303844451611142933424472212229343571119364146493835192034473122026293745174173439414626111420253136481011363846482618233234414649111214273448882021273648111114344042436915183435456115181929375151822314349269192024364810210111642432131736444548862830313448471172429344627   
rainbow lake Canada Member #25177 November 2, 2005 10764 Posts Offline  Posted: August 15, 2006, 8:32 am  IP Logged  
this is not printing out the right way?   
Solon, OH United States Member #3267 January 7, 2004 481 Posts Offline  Posted: August 15, 2006, 8:40 am  IP Logged  
LOL I hear math is the worst even though i got a B+ in algebra and statistics lol.   
rainbow lake Canada Member #25177 November 2, 2005 10764 Posts Offline  Posted: August 15, 2006, 8:56 am  IP Logged  
3102123334443 13152829334725 2131925323040 6262733414224 1316214246498 371538434732 269204344446 813212224394 13172133414238 62630384444516 1114293342447 221222934357 11193641464938 351920344731 2202629374517 4173439414626 11142025313648 10113638464826 182332344144649 1112142734488 8202127364811 111434042436 915183435456 115181929375 15182231434926 9192024364810 2101116424321 317364445488 6283031344847 1172429344627 Copy and paste did not work   
rainbow lake Canada Member #25177 November 2, 2005 10764 Posts Offline  Posted: August 15, 2006, 9:15 am  IP Logged  
if you read up and down you will notice only 8 of the first digits fall into the high number category. and 19 of the numbers fall into the odd category. that is first row up and down. there are only 7 numbers 6 or above first row. reading up and down. how about sums first 3 numbers across all the way down starts off as Sum 4,5,6,14,5,11,17,12,5,14,3,5,3,9,4,12,3,2,13,3,10,3,15,7,7,19,3,11,16,9 so root sums 4,5,6,5,5,2,8,3,5,5,3,5,3,9,4,3,3,2,4,3,1,3,6,7,7,1,3,2,7,9 so i will print another series number in 2 days or less. Can any one try and predict the next outcome. you can look at these number any way you like but best way to look at them is as 4 digits then 4 digits then 4 or 6 digits going across in the series. so one could predict the following as a next number 12192535464849 although if you read each digit in its corresponding spot up and down this may not be a good choice. if you care to try this I will tell you how much money you would have won if you played those numbers in a certain state, if this is too complicated then dont attempt, but it is worth a try because when this post is done it will open your eyes on how to chose numbers in a certain lottery that we all play very wisely.   
rainbow lake Canada Member #25177 November 2, 2005 10764 Posts Offline  Posted: August 15, 2006, 9:59 pm  IP Logged  
Ok looking in the first row my choice of first numbers would be , 2 or 5 or 8 or 4 or 9 for the next series draw. my second number will be a one so so far my combos are 21 51 81 41 91 but i will also have 10 combos starting with 10 11 12 13 14 15 16 17 18 19 now i will have to predict my 3rd row.   
rainbow lake Canada Member #25177 November 2, 2005 10764 Posts Offline  Posted: August 15, 2006, 10:20 pm  IP Logged  
One thing to keep in mind is on the first 3 digits that you chose will or can not be over sum of 18 so highest first 3 digits can be is 945 and 954 does not exist.   
rainbow lake Canada Member #25177 November 2, 2005 10764 Posts Offline  Posted: August 17, 2006, 12:24 am  IP Logged  
so i was off by a mile the next series number was 34825414822 Which if looking at it normal way would be 3,4,8,25,42,48, Bonus 22 I did manage to pull off a 4 number win using this system on the western 649 I had all digits but not on same line only ended up with a 4 hit on western i find its easier to look at the numbers drawn as a series such as 19141720388 and follow the columns up and down to predict next number to fall then move over to new column . the only reason this is in the pick 3 forumn was to give it exposure maybe thinking it should have been in the pick 6 games.   
