Approximating the Poisson Distribution (Edexcel IAL 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

Spec point — 3.2

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 $3.5\leq X_{N}<4.5$ 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 $P(3.5\leq X_{N}<4.5)=P(3.5<X_{N}<4.5)$

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

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

• $P(X\leq k)\approx P(X_{N}<k+0.5)$

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

• $P(X<k)\approx P(X_{N}<k-0.5)$

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

• $P(X\geq k)\approx P(X_{N}>k-0.5)$

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

• $P(X>k)\approx P(X_{N}>k+0.5)$

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

• For a closed inequality such as $P(a<X\leq b)$

  • Think about each inequality separately and use above
  • $P(X>a)\approx P(X_{N}>a+0.5)$
  • $P(X\leq b)\approx P(X_{N}>b+0.5)$
  • Combine to give
  • $P(a+0.5<X_{N}<b+0.5)$

Normal Approximation of Poisson

Spec point — 3.2

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

• A Poisson distribution $X \text{tilde} P o ( \lambda )$ can be approximated by a normal distribution $X_ N \text{tilde} N ( \mu , σ ² )$ provided

  • $λ$is large

• Remember that the mean and variance of a Poisson distribution are approximately equal, therefore the parameters of the approximating distribution will be:

  • $μ=λ$
  • $σ^{2}=λ$
  • $σ=\sqrt{λ}$

• 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

  • $μ=σ^{2}=λ$

• 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.

Answer:

Exam Hint

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