Properties of Poisson Distribution
What is a Poisson distribution?
- A Poisson distribution is a discrete probability distribution
- A discrete random variable follows a Poisson distribution if it counts the number of occurrences in a fixed time period given the following conditions:
- Occurrences are independent
- Occurrences occur at a uniform average rate for the time period (m)
- If follows a Poisson distribution then it is denoted
- m is the average rate of occurrences for the time period
- The formula for the probability of r occurrences is given by:
- for r = 0,1,2,...
- e is Euler’s constant 2.718...
- and
- There is no upper bound for the number of occurrences
- You will be expected to use the distribution function on your GDC to calculate probabilities with the Poisson distribution
What are the important properties of a Poisson distribution?
- The expected number (mean) of occurrences is m
- You are given this in the formula booklet
- The variance of the number of occurrences is m
- You are given this in the formula booklet
- Square root to get the standard deviation
- The mean and variance for a Poisson distribution are equal
- The distribution can be represented visually using a vertical line graph
- The graphs have tails to the right for all values of m
- As m gets larger the graph gets more symmetrical
- If and are independent then
- This extends to n independent Poisson distributions
- This extends to n independent Poisson distributions