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

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, X_N

  • For example, X = 4 for Poisson can be thought of as $3.5\le X_N<4.5$  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 $P(3.5\le X_N<4.5)=P(3.5<X_N<4.5)$

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

• $\mathrm{P}(X=k)\approx \mathrm{P}(k-0.5<X_N<k+0.5)$

• $\mathrm{P}(X\le k)\approx \mathrm{P}(X_N<k+0.5)$
  

  • You add 0.5 as you want to include $k$ in the inequality

• $\mathrm{P}(X<k)\approx \mathrm{P}(X_N<k-0.5)$

  • You subtract 0.5 as you don't want to include $k$ in the inequality

• $\mathrm{P}(X\ge k)\approx \mathrm{P}(X_N>k-0.5)$

  • You subtract 0.5 as you want to include $k$ in the inequality

• $\mathrm{P}(X>k)\approx \mathrm{P}(X_N>k+0.5)$

  • You add 0.5 as you don't want to include $k$ in the inequality

• For a closed inequality such as $\mathrm{P}(a<X\le b)$

  • Think about each inequality separately and use above

  • $\mathrm{P}(X>a)\approx \mathrm{P}(X_N>a+0.5)$

  • $\mathrm{P}(X\le b)\approx \mathrm{P}(X_N>b+0.5)$

  • Combine to give

  • $\mathrm{P}(a+0.5<X_N<b+0.5)$

Normal Approximation of Poisson

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

• A Poisson distribution $X~Po\left(\lambda \right)$  can be approximated by a normal distribution $X_N~N(\mu ,{\sigma }^2)$  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:

  • $\mu =\lambda$

  • ${\sigma }^2=\lambda$

  • $\sigma =\sqrt{\lambda }$

• 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

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

  • $\mu ={\sigma }^2=\lambda$

• 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