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

Revision Note

Author

Dan

Expertise

Maths

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   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  in the inequality
• You subtract 0.5 as you don't want to include  in the inequality
• You subtract 0.5 as you want to include  in the inequality
• You add 0.5 as you don't want to include  in the inequality
• For a closed inequality such as
• Think about each inequality separately and use above
• Combine to give

Normal Approximation of Poisson

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

• A Poisson distribution   can be approximated by a normal distribution  provided
• is large
• Remember that the mean and variance of a Poisson distribution are approximately equal, therefore the parameters of the approximating distribution will be:
• 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

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
• 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.

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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