The Poisson Probability Distribution Function, Mean and Variance

The random variable X is said to follow the Poisson probability distribution if it has the probability function

Example: A computer center manager, reports that his computer system experienced three component failures during the past 100 days.

a.    What is the probability of no failures in a given day?
b.    What is the probability of one or more component failures in a given day?
c.    What is the probability of at least two failures in a 3-day period?

Solution: A modern computer system has a very large number of components, each of which could fail and thus result in a computer system failure. To compute the probability of failures using the Poisson distribution, assume that all each of the millions of components has the same very small probability of failure. Also assume the first failure does not affect the probability of a second failure (in some cases, these assumptions may not hold, and more complex distributions would be used).

    
Example: Customers arrive at a photocopying machine at an average rate of two every 5 minutes. Assume that these arrivals are independent, with a constant arrival rate, and that this problem follows a Poisson model, with X denoting the number of arriving customers in a 5-minute period and mean  = 2. Find the probability that more than two customers arrive in a 5-minute period.

Solution: Since the mean number of arrivals in 5 minutes is two, then  = 2. To find the probability that more than two customers arrive, first compute the probability of at most two arrivals in a 5-minute period, and then use complement rule.