The central limit theorem also states that the sampling distribution will have the following properties: 1. The mean of the sampling distribution will be equal to the mean of population distribution: Central limit theorem states that sum of independent and identically distributed random numbers approach a Gaussian distribution. If $\{X_k\}_{k=1}^{n}$ are independent and identically distributed standard Cauchy random variables, then the sample mean $\left(\displaystyle \sum_{k=1}^n X_k \right)/n$ has the same standard Cauchy distribution. 6.2 The Central Limit Theorem Our objective is to show that the sum of independent random variables, when standardized, converges in distribution to the standard normal distribution. This highlights the importance of finite variance in the CLT. The Cauchy has another interesting property - the distribution of the sample average is that same as the distribution of an individual observation, so the scatter never diminishes, regardless of sample size. The distribution of an average tends to be Normal, even when the distribution from which the average is computed is decidedly non-Normal. Central limit theorem. In this article, we have discussed the central limit theorem, one of the most important results in statistics. Caveat: The Central Limit Theorem almost always holds, but caution is required in its application. The central limit theorem states that the sampling distribution of a sample mean is approximately normal if the sample size is large enough, even if the population distribution is not normal. Particularly, we emphasized understanding the key concept of “the sampling distribution of the mean”, whose distribution — not else — is guaranteed to follow the normal distribution … Normal Distribution. A random variable X is said to follow normal distribution with two parameters μ and σ and is denoted by X~N(μ, σ²). I have a simple question about CTL (Central limit theorem) Is it possible to use the Central limit theorem for standard Cauchy distribution? I think that it´s not possible because the Mean of the standard Cauchy distribution is undefined and the variance it´s the same. The normal distribution is also known as Gaussian distribution. Due to this theorem, this continuous probability distribution function is very popular and has several applications in variety of fields.