The Central Limit Theorem

According to The Weak Law of Large Numbers, the distribution of the Sample Mean

is concentrated near the mean μ of original distribution. In particular, its variance tends to zero, but the variance of:

$$S_n = X_1 + ... + X_n = n \bar X$$

increases to infinity and we cannot say that the distribution of \(S_n\) converges to anything. So, we need to scale it.

Consider \(S_n - nμ\) and scale it by a factor proportional to 1/\({\sqrt n}\). The Central Limit Theorem or CLT in short says that the distribution of this scaled random variable approaches a normal distribution.

Lets define \(Z_n\) as follows:

$$Z_n = {{S_n - nμ} \over {σ \sqrt n}}$$

hence, we can calculate its mean and variance by using basic theorems:

$$E[Z_n] = {{E[X_1 + X_2 + ... X_n] - E[nμ]} \over E[σ \sqrt n]}$$

$$= {{nμ - nμ} \over {σ \sqrt n}} = 0$$

$$Var(Z_n) = {Var(X_1 + X_2 + ... + X_n) \over σ^2n}$$

$$= {Var(X_1) + Var(X_2) + ... Var(X_n) \over σ^2n} = 1$$