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The Central Limit Theorem identifies the distribution of the sample mean and is arguably the most important theorem in probability theory. Let be a random variable, and let be a random sample for , such that each has a distribution identical to that of itself. Let be the sample mean; in other words, let be equal to . Because each is a random variable, is also a random variable. The Central Limit Theorem observes several important facts about the distribution of :

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  • Central Limit Theorem
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  • The Central Limit Theorem identifies the distribution of the sample mean and is arguably the most important theorem in probability theory. Let be a random variable, and let be a random sample for , such that each has a distribution identical to that of itself. Let be the sample mean; in other words, let be equal to . Because each is a random variable, is also a random variable. The Central Limit Theorem observes several important facts about the distribution of :
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  • The Central Limit Theorem identifies the distribution of the sample mean and is arguably the most important theorem in probability theory. Let be a random variable, and let be a random sample for , such that each has a distribution identical to that of itself. Let be the sample mean; in other words, let be equal to . Because each is a random variable, is also a random variable. The Central Limit Theorem observes several important facts about the distribution of : 1. * The distribution of is approximately normal, even when the underlying distribution is not. 2. * The expected value of the is equal to the expected value of . 3. * As the sample size increases, the variance of approaches zero.
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