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- This figure demonstrates the central limit theorem. The sample means are generated using a random number generator, which draws numbers between 0 and 100 from a uniform probability distribution. It illustrates that increasing sample sizes result in the 500 measured sample means being more closely distributed about the population mean . It also compares the observed distributions with the distributions that would be expected for a normalized Gaussian distribution, and shows the chi-squared values that quantify the goodness of the fit . The input into the normalized Gaussian function is the mean of sample means and the mean sample standard deviation divided by the square root of the sample size , which is called the standard deviation of the mean . (en)
- Comparison of probability density functions for the sum of fair 6-sided dice to show their convergence to a normal distribution with increasing , in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution . (en)
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| dbp:mathStatement
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- (en)
- , (en)
- . (en)
- and (en)
- Suppose is a sequence of independent random variables, each with finite expected value and variance (en)
- * .
Then
converges in distribution to . (en)
- Suppose is a sequence of i.i.d. random variables with and Then, as approaches infinity, the random variables converge in distribution to a normal : (en)
- Let be a random variable distributed uniformly on , and , where
* satisfy the lacunarity condition: there exists such that for all ,
* are such that (en)
- Define
If for some (en)
- and denotes the Euclidean norm on (en)
- where is a universal constant, (en)
- Suppose is a sequence of real-valued and strictly stationary random variables with for all (en)
- Let be independent -valued random vectors, each having mean zero. Write and assume is invertible. Let be a -dimensional Gaussian with the same mean and same covariance matrix as . Then for all convex sets (en)
- Construct
# If is absolutely convergent, , and then as where (en)
- # If in addition and converges in distribution to as then also converges in distribution to as (en)
- Let be independent, identically distributed random variables with and , and let be a sequence of non-negative integer-valued random variables that are independent of . Assume for each that and
where denotes convergence in distribution and is the normal distribution with mean 0, variance 1.
Then (en)
- Lyapunov’s condition
is satisfied, then a sum of converges in distribution to a standard normal random variable, as goes to infinity: (en)
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