Central Limit Theorem Proof
Let the independent random variables be X1, X2, X3, . . . . , Xn which are identically distributed and where their mean is zero(μ = 0) and their variance is one(σ2 = 1).
The Z score is given as, Z = [Tex]\dfrac{\overline X – \mu}{\frac{\sigma}{\sqrt n}} [/Tex]
Here, according to Central Limit Theorem, Z approximates to Normal Distribution as the value of n increases.
Let m(t) be the Moment Generating Function of Xi
⇒ M(0) = 1
⇒ M'(1) = E(Xi) = μ = 0
⇒ M”(0) = E(Xi2) = 1
The Moment Generating Function for Xi/√n is given as E[etXi/√n]
Since, X1 X2, X3 . . . Xn are independent, hence the Moment Generating Function for (X1 + X2 + X3 + . . . + Xn)/√n is given as [M(t/√n)]n
Let us assume as function
f(t) = log M(t)
⇒ f(0) = log M(0) = 0
⇒ f'(0) = M'(0)/M(0) = μ/1 = μ
⇒ f”(0) = (M(0).M”(0) – M'(0)2)/M'(0)2 = 1
Now, using L’ Hospital Rule we will find t/√n as t2/2
⇒ [M(t/√n)]2 = [ef(t/√n)]n
⇒ [enf(t/√n)] = e^(t2/2)
Thus the Central Limit Theorem has been proved by getting Moment Generating Function of a Standard Normal Distribution.
Central Limit Theorem
The Central Limit Theorem in Statistics states that as the sample size increases and its variance is finite, then the distribution of the sample mean approaches normal distribution irrespective of the shape of the population distribution. The query that how much the sample size should increase can be answered that if the sample size is greater than 30 then the statement of the Central Limit Theorem holds.
The central limit theorem posits that the distribution of sample means will invariably conform to a normal distribution provided the sample size is sufficiently large. This holds regardless of the underlying distribution of the population, be it normal, Poisson, binomial, or any alternative distribution.
In this article on the Central Limit Theorem, we will learn about the Central Limit Theorem definition, Central Limit Theorem examples, Central Limit Theorem Formulas, proof of the Central Limit Theorem, and Central Limit Theorem applications.
Table of Content
- Central Limit Theorem in Statistics
- Central Limit Theorem Definition
- Central Limit Theorem Formula
- Central Limit Theorem Proof
- Central Limit Theorem Examples
- Assumptions of the Central Limit Theorem
- Steps to Solve Problems on Central Limit Theorem
- Central Limit Theorem Applications
- Solved Examples on Central Limit Theorem
- Summary – Central Limit Theorem in Statistics