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Normal approximation to the binomial distribution

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Summary

Normal approximation to the binomial distribution

In a nutshell

Given certain specific conditions, it is possible to approximate a binomial distribution with a normal distribution.

Required conditions

Given $X \sim B(n,p)$ and $Y \sim N(\mu, \sigma^2)$ , the random variable $X$ can be approximated with $Y$ if:

$n$ is large.
$p$ is close to $0.5$ .

If $X \approx Y$ , then $\mu$ and $\sigma$ can be calculated using the following formulae:

$\boxed{\begin{aligned} \mu &= np\\ \sigma &= \sqrt{np(1-p)} \end{aligned}}$

Example 1

Given the random variable $X \sim B(250, 0.55)$ , explain why it is possible to approximate $X$ with a normal distribution, and calculate the values of $\mu$ and $\sigma^2$ if $X \approx Y \sim N(\mu, \sigma^2)$ .

$n$ is large and $p$ is close to $0.5$ . Therefore, it is possible to estimate $X$ with a normal distribution.

To calculate $\mu$ and $\sigma^2$ , use the formulae:

$\begin{aligned} \mu &= np\\&= 250 \times 0.55\\&=\underline{137.5}\\ \sigma &= \sqrt{np(1-p)}\\ \sigma^2&= \left( \sqrt{np(1-p)}\right)^2\\&= np(1-p)\\&=250\times 0.55 \times (1 - 0.55)\\&=\underline{61.875} \end{aligned}$

Calculating probabilities

To estimate probabilities of a binomial distribution using a normal distribution, a rounding error has to be applied due to the fact that the binomial distribution is a discrete distribution.

Example 2

The random variable $X \sim B(250, 0.55)$ is approximated with the random variable $Y \sim N(137.5, 61.875)$ . Estimate the following probabilities:

i) $P(X = 140)$ 

ii) $P(X \leq 180)$

iii) $P(130 \leq X \lt 150)$ 

Part i):

Use error intervals:

$\begin{aligned} P(X=140)& \approx P(139.5 \lt Y \lt 140.5)\\ &\approx \underline{0.0482} \end{aligned}$ 

Part ii):

Since $180$ is included in the inequality, the upper bound is $180.5$ .

$\begin{aligned} P(X \leq 180) &\approx P(Y \lt 180.5)\\ &\approx \underline{1} \end{aligned}$ 

Part iii):

$\begin{aligned} P(130 \leq X \lt 150) &\approx P(129.5 \lt Y \lt 149.5)\\ &\approx \underline{0.7819} \end{aligned}$ 

Learn with Basics

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Basics of probability

The binomial distribution

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