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Join date: May 18, 2022

## Central Limit Theorem

The Central Limit Theorem (CLT) states that the Central Limit Theorem (CLT) states that the sum of a large number of independent and identically distributed (IID) random variables drawn from a certain distribution converges to a Gaussian, with mean 0 and variance 1, under certain conditions. The Gaussian distribution is the most symmetric distribution; the average of many values can be approximated by the mean and the standard deviation of the distribution. There is a resemblance between the Central Limit Theorem and the Law of Large Numbers (LLN) that approximates the average of a large number of observations by the average of a sample. They can be used to calculate the average of a large number of observations. The Central Limit Theorem allows you to find the normal distribution of the sum of independent and identically distributed random variables. The Central Limit Theorem (CLT) can be stated as follows: If X1, X2, X3, …, XN are independent and identically distributed random variables having a common probability distribution and X = X1 + X2 + X3 + … + XN, then X has a normal distribution. For a detailed proof of the Central Limit Theorem, please visit: The Central Limit Theorem follows from the Law of Large Numbers, which states that if X1, X2, X3, …, XN are independent random variables drawn from the same probability distribution, the average of N such random variables also follows a normal distribution (the Central Limit Theorem is a special case of the Law of Large Numbers). These two results, together, lead to a practical method of calculating the properties of the sum of independent random variables. Uses of Central Limit Theorem: In practice, we often have data that we would like to analyze and where the data may have a non-normal distribution (for example, data that is not uniformly distributed), but where the central limit theorem is still applicable. The Central Limit Theorem is useful in such cases because it allows the derivation of statistical properties for large numbers of random variables, independent of their individual properties. Limitations of Central Limit Theorem: The Central Limit Theorem is only applicable when the random variables are "random" and do not depend on previous observations. For example, if your test questions were distributed among all students, then the test would be random, but the previous answers would not