Philosophical question about central limit theorem

In summary, the conversation discusses the central limit theorem and its application to a distribution with mean 0 and cumulative distribution of 0 for all values less than -1. The fallacy in the paradox arises from the incorrect application of the theorem, where the normalization involves multiplying the sample mean by the square root of n. The conversation concludes by suggesting to use statistical software to simulate random variables and graph the results to better understand the application of the central limit theorem.
  • #1
coquelicot
299
67
Well, this is probably a stupid question, but I don't see why (yet).

Let Xi be random variables identically distributed, with mean 0, such that the cumulative distribution is = 0 for all -1 < x < 1. So, I believe it is clear that for all n, the cumulative distribution of Z = (X1 + X2 ... Xn)/n is = 0 for all x < -1. But the central limit theorem implies that this distribution (normalized by the square root of n), converges in distribution to the Normal distribution. So, for some n sufficiently large, the cumulative distribution of Z must be > 0 for some x < -1. Where is the fallacy in this paradox?
thx.
 
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  • #2
coquelicot said:
But the central limit theorem implies that this distribution (normalized by the square root of n),
.

The normalization involves multiplying the sample mean by [itex] \sqrt{n} [/itex]. The quantity that is approximately normally distributed (for a sample of size n , sample mean [itex] S_n [/itex] and population mean [itex] \mu [/itex] for each individual random variable) is [itex] \sqrt{n}(S_n - \mu) [/itex].
 
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  • #3
If you have issues with the application of the theorem to a particular distribution I would recommend opening a statistical software package and simulating a number of random variables and then graphing the result of the random vectors that get the arithmetic mean and see what happens.

It should help convince you through some examples of how the CLT holds.
 
  • #4
Thx all.
I have understood my error.
 
  • #5


I appreciate your curiosity and willingness to question assumptions. The central limit theorem is a powerful tool in statistics that allows us to make predictions about the behavior of a large number of random variables. However, it is important to understand the assumptions and limitations of this theorem.

In this case, the fallacy lies in assuming that the cumulative distribution of Z will always be equal to 0 for all x < -1. This is not necessarily true, as the central limit theorem only guarantees that the distribution of Z will approach the Normal distribution as n becomes large. This means that for smaller values of n, the distribution of Z may not follow the Normal distribution and may have non-zero values for x < -1.

Additionally, the assumption that the random variables Xi are identically distributed with mean 0 may also be a limitation. The central limit theorem requires that the random variables have finite variance, so if the variance of Xi is very small, it may take a larger value of n for the distribution of Z to approach the Normal distribution.

In conclusion, while the central limit theorem is a powerful tool, it is important to understand its assumptions and limitations. It is always good to question and explore paradoxes, but it is also important to carefully examine the underlying assumptions and data before drawing conclusions.
 

1. What is the central limit theorem?

The central limit theorem is a mathematical concept in statistics that states that when a large number of independent random variables are added together, their sum will tend towards a normal distribution, regardless of the distribution of the individual variables. In simpler terms, it explains why many natural phenomena tend to follow a bell-shaped curve.

2. Why is the central limit theorem important?

The central limit theorem is important because it allows us to make statistical inferences about a population based on a sample, as long as the sample is large enough. This concept is the foundation of many statistical techniques and plays a crucial role in data analysis and decision making.

3. Are there any limitations to the central limit theorem?

Yes, there are some limitations to the central limit theorem. The theorem assumes that the sample is a random sample from a population and that the samples are independent of each other. It also requires a sufficiently large sample size to accurately approximate a normal distribution. Violating these assumptions can lead to inaccurate results.

4. How is the central limit theorem used in real life?

The central limit theorem is used in various fields such as finance, economics, psychology, and physics to analyze and interpret data. For example, in finance, it is used to predict stock prices, in economics to estimate demand and supply, and in psychology to study behavior patterns. It is also used in quality control and product testing to ensure that products meet certain standards.

5. Can the central limit theorem be proven?

No, the central limit theorem cannot be proven as it is a mathematical theorem based on assumptions. However, it has been extensively tested and has been found to hold true under various conditions. It is widely accepted as a fundamental concept in statistics and has been used successfully in various applications.

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