The Central Limit Theorem (CLT) is a fundamental concept in probability theory and statistics. It states that if you take a large enough sample of independent and identically distributed random variables from any distribution with a finite mean and variance, then the sample mean of those variables will approach a normal distribution as the sample size increases, regardless of the underlying distribution.
In other words, the CLT tells us that the sum of a large number of random variables will tend to be normally distributed, even if the original random variables themselves are not normally distributed. This is because the normal distribution is a stable distribution, meaning that it is very resistant to changes in the underlying distribution of the random variables being averaged.
The CLT has important applications in many areas of science and engineering, particularly in statistics and data analysis. It allows us to make predictions about the distribution of a population based on a sample of that population, and it is used extensively in hypothesis testing, confidence interval estimation, and other statistical procedures.
Overall, the Central Limit Theorem is a powerful tool for understanding the behavior of random variables and the relationship between sample statistics and population parameters.