Binomial Distribution Approximated by Normal Distribution

Theorem

Let $X$ be a discrete random variable which has the binomial distribution $\Binomial n p$.

Then for large $n$ and such that both $n p$ and $n q$ are approximately $5$ or more:

$\Binomial n p \approx \Gaussian {n p} {n p q}$

where $\Gaussian {n p} {n p q}$ denotes the normal distribution.


Proof


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Sources

  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): binomial distribution
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): binomial distribution
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): normal approximation
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