📂Probability Distribution

Derivation of the Standard Normal Distribution as a Limiting Distribution of the Binomial Distribution

Theorem

De Moivre-Laplace Theorem

If $X_i \sim B(1,p)$ and $Y_n = X_1 + X_2 + \cdots + X_n$, then $Y_n \sim B(n,p)$ and $${ { Y_n - np } \over {\sqrt{ np(1-p) } } }\overset{D}{\to} N(0,1)$$

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• $N \left( \mu , \sigma^{2} \right)$ is a normal distribution with mean $\mu$ and variance $\sigma^{2}$.
• $B(n,p)$ is a binomial distribution with $n$ trials and probability $p$.
• $\overset{D}{\to}$ denotes convergence in distribution.

Description

This theorem is also known as the De Moivre–Laplace Theorem, and is widely known as a special case of the central limit theorem.

From the beginning of learning statistics, it has been taught that as the sample size of a binomial distribution increases, it approximates a normal distribution. This is evident from experience, and the process of proof does not hold great significance, but it serves as a good example to concretely grasp convergence in distribution, which may not be intuitively obvious from formulas alone.

Derivation

$${ { Y_n - np } \over {\sqrt{ np(1-p) } } } = \sqrt{n} { { \overline{X_n} - p } \over { \sqrt{p(1-p)} } }$$ Since $X_i \sim B(1,p)$, we have $E(X_i ) = p$ and $\Var(X_i ) = p(1-p)$. Furthermore, by the central limit theorem, $$\sqrt{n} { { \overline{X_n} - p } \over { \sqrt{p(1-p)} } } \overset{D}{\to} N(0,1)$$

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