Central Limit Theorem:

From a practical point of view, the most useful limit theorem in probability theory as the central limited theorem that permits us to approximate the sampling distribution of the sample mean x¯ of a random sample X of size n from a great population by an approximate sample distribution when n is sufficiently large. Recall that the sample distribution of k for sample from a normal population is known to be normal. The central limit theorem asserts that, under certain conditions, the distribution of X is approximately normal even when the population from which sample is drawn, is not normal. The importance of the theorem is due to the fact that most of the statistical procedures used to draw statistical inference, are based on the distribution of a sample mean and that the exact distribution of a sample mean is difficult to obtain in many cases.

The central limit theorem is stated and proved under many conditions and assumptions. We state the theorem, without giving a proof, under a very restrictive condition. However, the statement is satisfactory for our purpose of using it in practice.