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The Central Limit Theorem states that the distribution of the sum (or average) of a large number of independent, identically distributed random variables approaches a normal distribution as the sample size goes to infinity, regardless of the underlying distribution of the individual random variables. The CLT is concerned with the limiting distribution.

The Central Limit Theorem (CLT) is a fundamental result in probability theory and statistics. It describes the behavior of the sum (or average) of a large number of independent, identically distributed (i.i.d.) random variables. The theorem states that regardless of the distribution of the individual random variables, the distribution of the sample mean approaches a normal (Gaussian) distribution as the sample size becomes large. This has significant implications for statistical inference and hypothesis testing.

The Central Limit Theorem can be stated as follows:

Central Limit Theorem (CLT): Let X₁, X₂, ..., Xₙ be independent and identically distributed random variables with mean (expected value) μ and variance σ². If n is sufficiently large, the distribution of the sample mean (X̄) approaches a normal distribution with mean μ and variance σ²/n:

          X̄ ~ N(μ, σ²/n) ----------------------------------------------- [3958a]

Here are some key points and details about the CLT:

1. Independence and Identical Distribution: The random variables X₁, X₂, ..., Xₙ must be independent of each other and have the same probability distribution. This is a crucial assumption for the CLT to hold.

2. Mean and Variance: The random variables should have a well-defined mean (μ) and variance (σ²). These parameters describe the center and spread of the distribution, respectively.

3. Sample Size (n): The CLT does not specify a precise threshold for the sample size to be considered "sufficiently large." However, as a rule of thumb, n is often considered large enough when n ≥ 30. In practice, the validity of the CLT depends on the specific distribution of the random variables and the desired level of approximation.

4. Approximation: The CLT states that the distribution of the sample mean becomes approximately normal as n increases. This approximation improves as n gets larger.

Now, let's illustrate the CLT with equations:

Suppose you have n i.i.d. random variables, X₁, X₂, ..., Xₙ, each with mean μ and variance σ². The sample mean X̄ is defined as:

          X̄ = (X₁ + X₂ + ... + Xₙ)/n ----------------------------------------------- [3958b]

The CLT states that, as n becomes large:

          X̄ ~ N(μ, σ²/n) ----------------------------------------------- [3958c]

In this equation:

• X̄ is the sample mean.
• N(μ, σ²/n) represents a normal distribution with mean μ and variance σ²/n.

As an example, if you were conducting a large number of trials (n) of a random experiment and calculating the average outcome each time, the distribution of those sample means would approach a normal distribution with the specified mean and variance as n increases.

The CLT has far-reaching applications in statistics because it allows us to make probabilistic statements about sample means, even when we do not know the underlying distribution of the random variables, as long as we have a sufficiently large sample size. This is particularly important in hypothesis testing and confidence interval estimation.

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