Central limit theorem
A core theorem of probability theory stating that the mean of a large number of independent and identically distributed random variables approximately follows a normal distribution, forming the theoretical basis of statistical inference.
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By the central limit theorem, √n(Ȳ_n − σ²) converges in distribution to N(0, τ)
Applying the law of large numbers requires iid and finite mean, while applying the central limit theorem additionally requires finite variance
The Cauchy distribution is a special case of the t-distribution with degrees of freedom ν=1; its expected value and variance do not exist, the sample mean does not converge to a stable value, and the CLT fails in this case.
The central limit theorem states that the standardized mean of independent and identically distributed random variables (with finite variance) approaches a standard normal distribution as the number of variables grows large enough, regardless of the type of the original distribution.
The ubiquity of the Gaussian distribution stems from the central limit theorem, which states that the sum of a large number of independent random variables tends toward a normal distribution.
All Facts (5)
By the central limit theorem, √n(Ȳ_n − σ²) converges in distribution to N(0, τ)
50%UnverifiedApplying the law of large numbers requires iid and finite mean, while applying the central limit theorem additionally requires finite variance
50%UnverifiedThe Cauchy distribution is a special case of the t-distribution with degrees of freedom ν=1; its expected value and variance do not exist, the sample mean does not converge to a stable value, and the CLT fails in this case.
50%UnverifiedThe central limit theorem states that the standardized mean of independent and identically distributed random variables (with finite variance) approaches a standard normal distribution as the number of variables grows large enough, regardless of the type of the original distribution.
50%UnverifiedThe ubiquity of the Gaussian distribution stems from the central limit theorem, which states that the sum of a large number of independent random variables tends toward a normal distribution.
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