Terms you'll find helpful in completing today's challenge are outlined below.

Poisson Random Variables

We've already learned that we can break many problems down into terms of , , and and use the following formula for binomial random variables:

But what do we do when cannot be calculated using that formula? Enter the Poisson random variable.

Poisson Experiment

A Poisson experiment is a statistical experiment that has the following properties:

• The outcome of each trial is either success or failure.
• The average number of successes that occurs in a specified region is known.
• The probability that a success will occur is proportional to the size of the region.
• The probability that a success will occur in an extremely small region is virtually zero.
  
  

Poisson Distribution

A Poisson random variable is the number of successes that result from a Poisson experiment. The probability distribution of a Poisson random variable is called a Poisson distribution:

• is the average number of successes that occur in a specified region.
• is the actual number of successes that occur in a specified region.
• is the Poisson probability, which is the probability of getting exactly successes when the average number of successes is .
  
  

Example

Acme Realty company sells an average of homes per day. What is the probability that exactly homes will be sold tomorrow?

Here, and , so

Example

Suppose the average number of lions seen by tourists on a one-day safari is . What is the probability that tourists will see fewer than lions on the next one-day safari?

Special Case

Consider some Poisson random variable, . Let be the expectation of . Find the value of .

Let be the variance of . Recall that if a random variable has a Poisson distribution, then:

• 
  
  Now, we'll use the following property of expectation and variance for any random variable, :
  So, for any random variable having a Poisson distribution, the above result can be rewritten as: