When can I use a normal distribution to approximate a Poisson distribution?
- A Poisson distribution can be approximated by a normal distribution provided
- is sufficiently large ( > 15)
- 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
What are continuity corrections?
- The Poisson 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 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
Do I need to use continuity corrections?
- As the Poisson distribution X is discrete and normal distribution XN is continuous you will need to use continuity corrections
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 (approximately > 15) then it will be a close approximation
The number of hits on a revision web page per hour can be modelled by the Poisson distribution with a mean of 40. Find the probability that there are more than 50 hits on the webpage in a given hour.