Approximating the Poisson Distribution (Edexcel International A Level Maths: Statistics 2)

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Calculating probabilities using a binomial or Poisson distribution can take a while. Under certain conditions we can use a normal distribution to approximate these probabilities. As we are going from a discrete distribution (binomial or Poisson) to a continuous distribution (normal) we need to apply continuity corrections.

Continuity Corrections

What are continuity corrections?

  • The binomial and Poisson distribution are 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 Poisson can be thought of as 3.5 less or equal than X subscript N less than 4.5  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 P left parenthesis 3.5 less or equal than X subscript N less than 4.5 right parenthesis equals P left parenthesis 3.5 less than X subscript N less than 4.5 right parenthesis

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
  • straight P left parenthesis X equals k right parenthesis almost equal to straight P left parenthesis k minus 0.5 less than X subscript N less than k plus 0.5 right parenthesis
  • straight P left parenthesis X less or equal than k right parenthesis almost equal to straight P left parenthesis X subscript N less than k plus 0.5 right parenthesis
    • You add 0.5 as you want to include k in the inequality
  • straight P left parenthesis X less than k right parenthesis almost equal to straight P left parenthesis X subscript N less than k minus 0.5 right parenthesis
    • You subtract 0.5 as you don't want to include k in the inequality
  • straight P left parenthesis X greater or equal than k right parenthesis almost equal to straight P left parenthesis X subscript N greater than k minus 0.5 right parenthesis
    • You subtract 0.5 as you want to include k in the inequality
  • straight P left parenthesis X greater than k right parenthesis almost equal to straight P left parenthesis X subscript N greater than k plus 0.5 right parenthesis
    • You add 0.5 as you don't want to include k in the inequality
  • For a closed inequality such as straight P left parenthesis a less than X less or equal than b right parenthesis
    • Think about each inequality separately and use above
    • straight P left parenthesis X greater than a right parenthesis almost equal to straight P left parenthesis X subscript N greater than a plus 0.5 right parenthesis
    • straight P left parenthesis X less or equal than b right parenthesis almost equal to straight P left parenthesis X subscript N greater than b plus 0.5 right parenthesis
    • Combine to give
    • straight P left parenthesis a plus 0.5 less than X subscript N less than b plus 0.5 right parenthesis

Normal Approximation of Poisson

When can I use a normal distribution to approximate a Poisson distribution?

  • A Poisson distribution X tilde P o left parenthesis lambda right parenthesis  can be approximated by a normal distribution X subscript N tilde N left parenthesis mu comma sigma squared right parenthesis  provided
    • lambda is large
  • Remember that the mean and variance of a Poisson distribution are approximately equal, therefore the parameters of the approximating distribution will be:
    • mu equals lambda
    • sigma squared equals lambda
    • sigma equals square root of lambda
  • The greater the value of λ in a Poisson distribution, the more symmetrical the distribution becomes and the closer it resembles the bell-shaped curve of a normal distribution

2-4-2-approximations-of-distributions-diagram-1

Why do we use approximations?

  • If there are a large number of values for a Poisson distribution there could be a lot of calculations involved and it is inefficient to work with the Poisson distribution
    • These days calculators can find Poisson probabilities so approximations are no longer necessary
    • However it can still be 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 Poisson distribution function)

How do I approximate a probability?

  • STEP 1: Find the mean and variance of the approximating distribution
      • mu equals sigma squared equals lambda
  • STEP 2: Apply continuity corrections to the inequality
  • STEP 3: Find the probability of the new corrected inequality
      • Find the standard normal probability and use the table of the normal distribution
  • The probability will not be exact as it is an approximation but provided λ is large enough then it will be a close approximation

Worked example

The number of hits on a revision web page per hour can be modelled by the Poisson distribution with a mean of 40.  Use a normal approximation to find the probability that there are more than 50 hits on the webpage in a given hour.

2-4-2-approximations-of-distributions-we-solution-1

Exam Tip

  • The question will make it clear if an approximation is to be used, λ will be bigger than the values in the formula booklet.

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Dan

Author: Dan

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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