Central Limit Theorem (CLT) - Python for Integrated Circuits - - An Online Book - |
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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:
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:
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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