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Talk:Law of Large Numbers (view source)
Revision as of 03:06, March 15, 2012
, 03:06, March 15, 2012add my explanation of why the counterexample is not really a good one
:The Theory of Evolution, as it is commonly taught, does claim that some new species arose due to random fluctuations.
:The Theory of Evolution, as it is commonly taught, does claim that some new species arose due to random fluctuations.
::Man. This is so wrong I don't even know where to start. So I'm not going to because I'll get 90/10'ed out the door. It's like an infinite manifold of wrongness. Just know this, you are applying probability theory -a theory you only partially understand- to evolution, a theory you completely don't understand. --[[User:JoshuaB|JoshuaB]] 20:48, 14 March 2012 (EDT)
::Man. This is so wrong I don't even know where to start. So I'm not going to because I'll get 90/10'ed out the door. It's like an infinite manifold of wrongness. Just know this, you are applying probability theory -a theory you only partially understand- to evolution, a theory you completely don't understand. --[[User:JoshuaB|JoshuaB]] 20:48, 14 March 2012 (EDT)
I will explain why I believe that the law of large numbers is not a good (and probably, to be honest, misleading) counterexample to evolution:
Evolutionists would argue that the hypotheses of the law of large numbers (that the random variables are independent and identically distributed) are not satisfied. I understand that the concept of i.i.d. is technical and difficult to explain, but it is nevertheless essential to the theorem. As an example, take the sample space <math>\{H, T\}</math> corresponding to flipping a fair coin once, and for each positive integer <math>i</math>, let <math>X_i</math> equal 1 if the outcome is <math>H</math> and 0 if the outcome is <math>T</math>. In other words, if the outcome is <math>H</math>, then <math>X_1 = X_2 = \cdots = 1</math>, and if the outcome is <math>T</math>, then <math>X_1 = X_2 = \cdots =0</math>. Note that <math>X_1, X_2, \ldots</math> are all Bernoulli random variables with parameter <math>p=\frac{1}{2}</math>, so they are identically distributed. However, it is clear that the law of large numbers does not hold, as <math>\frac{X_1 + \cdots + X_n}{n}</math> is always either 1 or 0. Therefore, independence is an essential hypothesis. In general, time series are not independent (imagine the Dow Jones index where the closing values are always independent and seeing the index go from 13000 one day to 175 the next!).
Random variables must be functions to the real numbers, so it is ambiguous what the statement "populations of species can randomly drift away from their 'mean'" is intended to mean without a quantification. Nevertheless, using height as a characteristic to measure for the sake of argument, it is clear why the law of large numbers does not hold. If we observe the height of a randomly selected member of a population each year, the observations are random variables. Nevertheless, evolutionists would provide two explanations for why the law of large numbers does not hold in this experiment. First, evolutionists would argue that factors like [[genetic drift]] allow for the events of one time period to affect the height distribution of a population in future time periods. Second, evolutionists would suggest that trends in random height suggest precisely the conclusion that the distribution of population heights is changing with each year, indicating that the random variables are not identically distributed. By the way, looking at the statement of the weak law of large numbers, it is clear that if the random variables are not identically distributed (especially if they do not have the same mean), then the "mean" <math>\mu</math> in the statement of the theorem is completely meaningless, mathematically. [[User:GregG|GregG]] 23:06, 14 March 2012 (EDT)