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.