# Edexcel A Level Maths: Statistics:复习笔记4.4.2 Normal Approximation of Binomial

### Normal Approximation of Binomial

#### When can I use a normal distribution to approximate a binomial distribution?

• A binomial distribution  can be approximated by a normal distribution   provided
• n is large
• p is close to 0.5
• The mean and variance of a binomial distribution can be calculated by:

#### Why do we use approximations?

• These days calculators can calculate binomial probabilities so approximations are no longer necessary
• However it is easier to work with a normal distribution
• You can calculate the probability of a range of values quickly
• You can use the inverse normal distribution function (most calculators don't have an inverse binomial distribution function)

#### What are continuity corrections?

• The binomial distribution is discrete and the normal distribution is continuous
• A continuity correction takes this into account when using a normal approximation
• The probability being found will need to be changed from a discrete variable, X,   to a continuous variable, XN
• For example, X = 4 for binomial can be thought of as  for normal as every number within this interval rounds to 4
• Remember that for a normal distribution the probability of a single value is zero so

#### How do I apply continuity corrections?

• Think about what is largest/smallest integer that can be included in the inequality for the discrete distribution and then find its upper/lower bound
• You add 0.5 as you want to include k in the inequality
• You subtract 0.5 as you don't want to include k in the inequality
• You subtract 0.5 as you want to include k in the inequality
• You add 0.5 as you don't want to include k  in the inequality
• For a closed inequality such as
• Think about each inequality separately and use above
• Combine to give

#### How do I approximate a probability?

• STEP 1: Find the mean and variance of the approximating distribution
• STEP 2: Apply continuity corrections to the inequality
• STEP 3: Find the probability of the new corrected inequality
• Use the "Normal Cumulative Distribution" function on your calculator
• The probability will not be exact as it is an approximate but provided n is large and p is close to 0.5 then it will be a close approximation

#### Worked Example

The random variable .

Use a suitable approximating distribution to approximate .

#### Exam Tip

• In the exam, only use a normal approximation if the question tells you to. Otherwise use the binomial distribution.