This tag should be used for each question where the term "central limit theorem" and with the tag (tag:probability-limit-theorems). The central limit theorem states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough.

The **central limit theorem** (CLT) is one of the most important results in probability theory. It states that,

Let $~X_1,~X_2,\cdots,~X_n~$ be Independent and identically distributed (i.i.d.) random variables with expected value $~E~X_i=μ<∞~$ and variance $~0<Var(X_i)=σ^2<∞~$. Then, the random variable $$\begin{align}%\label{} Z_{\large n}=\frac{\overline{X}-\mu}{\sigma / \sqrt{n}}=\frac{X_1+X_2+...+X_{\large n}-n\mu}{\sqrt{n} \sigma} \end{align}$$ converges in distribution to the standard normal random variable as $~n~$ goes to infinity, that is $$\begin{align}%\label{} \lim_{n \rightarrow \infty} P(Z_{\large n} \leq x)=\Phi(x), \qquad \textrm{ for all }x \in \mathbb{R}, \end{align}$$where $~\Phi(x)~$ is the standard normal CDF.

**How to Apply The Central Limit Theorem :**

- Write the random variable of interest, $~Y~$, as the sum of $~n~$ i.i.d. random variable $~X_i~$'s: $$Y=X_1+X_2+\cdots+X_n$$
- Find $~E~Y~$ and $~Var(Y)~$ by noting that $$E~Y=nμ,\qquad Var(Y)=n~σ^2,$$ where $~μ=E~X_i~$ and $~σ^2=Var(X_i)~$.
- According to the CLT, conclude that $$\frac{Y−E~Y}{\sqrt{Var(Y)}}=\frac{Y−nμ}{\sqrt n~σ}$$ is approximately standard normal; thus, to find $~P(y_1≤Y≤y_2)~$, we can write $$P(y_1≤Y≤y_2)=P\left(\frac{y_1−nμ}{\sqrt n~σ}≤\frac{Y−nμ}{\sqrt n~σ}≤\frac{y_2−nμ}{\sqrt n~√σ}\right)$$ $$≈\phi\left(\frac{y_2−nμ}{\sqrt n~σ}\right)−\phi\left(\frac{y_1−nμ}{\sqrt n~σ}\right).$$

**References:**

https://en.wikipedia.org/wiki/Central_limit_theorem

http://mathworld.wolfram.com/CentralLimitTheorem.html

https://www.analyticsvidhya.com/blog/2019/05/statistics-101-introduction-central-limit-theorem/