Subject: Probability & Statistics
Help! I am a secondary education mathematics teacher with a probability question:
A skeptic gives the following argument to show that there must be a flaw in the central limit theorem: We know that the sum of independent Poisson random variables follows a Poisson distribution with a parameter that is the sum of the parameters of the summands. In particular, if n independent Poisson random variables, each with parameter 1/n, are summed, the sum has a Poisson distribution with parameter 1. The central limit theoren says the sum tends to a normal distribution, but Poisson distribution with parameter 1 is not normal.
What do you think of this argument?
Thanks for your input.
The Central Limit Theorem says that for any given distribution (with finite variance), there exists an N such that the sum of N independent copies of random variables from this distribution is "approximately normal". It is important that the size of N depends on the particular distribution. For a Poisson distribution with parameter 1, you get approximate normality with N=30, but for a Poisson distribution with parameter 1/100, you would need N=3000. In your example for a Poisson distribution with parameter 1/n you get approximate normality with N=30n.
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