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A Poisson probability distribution of a discrete random variable gives the probability of a number of events occurring in a fixed interval of time or space, if these events happen at a known average rate and independently of the time since the last event. The Poisson distribution may be used to approximate the binomial, if the probability of success is "small" (less than or equal to 0.01) and the number of trials is "large" (greater than or equal to 25). Other rules of thumb are also suggested by different authors, but all recognize that the Poisson distribution is the limiting distribution of the binomial as n increases and p approaches zero.
The formula for computing probabilities that are from a Poisson process is:
where P(X) is the probability of successes, μ (pronounced mu) is the expected number of successes, e is the natural logarithm approximately equal to 2.718, and X is the number of successes per unit, usually per unit of time.
X ~ P ( μ ) means that X has a Poisson probability distribution where X = the number of occurrences in the interval of interest.
X takes on the values x = 0, 1, 2, 3, ...
The mean μ or λ is typically given.
The variance is
σ
^{2} =
μ , and the standard deviation is
$\sigma \text{=}\sqrt{\mu}$ .
When P ( μ ) is used to approximate a binomial distribution, μ = np where n represents the number of independent trials and p represents the probability of success in a single trial.
Use the following information to answer the next six exercises: On average, a clothing store gets 120 customers per day.
Assume the event occurs independently in any given day. Define the random variable X .
What values does X take on?
0, 1, 2, 3, 4, …
What is the probability of getting 150 customers in one day?
What is the probability of getting 35 customers in the first four hours? Assume the store is open 12 hours each day.
0.0485
What is the probability that the store will have more than 12 customers in the first hour?
What is the probability that the store will have fewer than 12 customers in the first two hours?
0.0214
Which type of distribution can the Poisson model be used to approximate? When would you do this?
Use the following information to answer the next six exercises: On average, eight teens in the U.S. die from motor vehicle injuries per day. As a result, states across the country are debating raising the driving age.
Assume the event occurs independently in any given day. In words, define the random variable X .
X = the number of U.S. teens who die from motor vehicle injuries per day.
X ~ _____(_____,_____)
What values does X take on?
0, 1, 2, 3, 4, ...
For the given values of the random variable X , fill in the corresponding probabilities.
Is it likely that there will be no teens killed from motor vehicle injuries on any given day in the U.S? Justify your answer numerically.
No
Is it likely that there will be more than 20 teens killed from motor vehicle injuries on any given day in the U.S.? Justify your answer numerically.
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