*Simillar Question

State central limit theorem with its importance

The Central Limit Theorem (Liapounoff’s Form)

    If $X_1,X_2,X_3…..X_n…..$ be a sequence of independent identically distributed RVs with $E(X_i )=μ_i$ and $Var(X_i )=σ_i^2, i=1,2….$ and if $S_n=X_1+X_2+⋯.X_n$ then under certain general conditions, $S_n$ follows a normal distribution with mean $μ=∑_{i=1}^n μ_i$ and variance $σ^2=∑_{i=1}^n σ_i^2$ as n tends to infinity.

Central Limit Theorem (Lindberg-Levy’s Form)

If $X_1,X_2,X_3…..X_n…..$ be a sequence of independent identically distributed RVs with E($X_i )$=μ and Var$(X_i )=σ^2$, i=1,2…. and if $S_n=X_1+X_2+⋯.X_n$ then under certain general conditions, $S_n$ follows a normal distribution with mean nμ and variance n$σ^2$ as n tends to infinity.

Corollary :

If $\bar{X }=1/n(X_1+X_2+⋯.X_n)$ then E($\bar{X}$)=$\frac{1}n nμ=μ $and Var$(\bar{X })$=$\frac1{n^2} .nσ^2=\frac{σ^2}n$

∴ $\bar{X} $ follows $N(μ,\fracσ{( \sqrt n )})$ as n→∞

The importance of the central limit theorem stems from the fact that, in many real applications, a certain random variable of interest is a sum of a large number of independent random variables. In these situations, we are often able to use the CLT to justify using the normal distribution. Examples of such random variables are found in almost every discipline. Here are a few:

Laboratory measurement errors are usually modeled by normal random variables.

• In communication and signal processing, Gaussian noise is the most frequently used model for noise.

• In finance, the percentage changes in the prices of some assets are sometimes modeled by normal random variables.

• When we do random sampling from a population to obtain statistical knowledge about the population, we often model the resulting quantity as a normal random variable.

The CLT is also very useful in the sense that it can simplify our computations significantly. If we have a problem in which we are interested in a sum of one thousand i.i.d. random variables, it might be extremely difficult, if not impossible, to find the distribution of the sum by direct calculation. Using the CLT we can immediately write the distribution, if we know the mean and variance of the Xi's.

The value of n generally depends on the distribution of the Xis. Nevertheless, as a rule of thumb it is often stated that if n is larger than or equal to 30, then the normal approximation is very good