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# Poisson Random Variable and Probability Formula for Beginning Statistics

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By — McGraw-Hill Professional
Updated on Aug 12, 2011

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### Poisson Random Variable

The binomial random variable is applicable when counting the number of occurrences of an event called success in a finite number of trials. When the number of trials is large or potentially infinite, another random variable called the Poisson random variable may be appropriate. The Poisson probability distribution is applied to experiments with random and independent occurrences of an event. The occurrences are considered with respect to a time interval, a length interval, a fixed area, or a particular volume.

EXAMPLE 5.21 The number of calls to arrive per hour at the reservation desk for Regional Airlines is a Poisson random variable. The calls arrive randomly and independently of one another. The random variable X is defined to be the number of calls arriving during a time interval equal to one hour and may be any number from 0 to some very large value.

EXAMPLE 5.22 The number of defects in a 10-foot coil of wire is a Poisson random variable. The defects occur randomly and independently of one another. The random variable X is defined to be the number of defects in a 10-foot coil of wire and may be any number between 0 and some very large value. The interval in this example is a length interval of 10 feet.

EXAMPLE 5.23 The number of pinholes in 1-yd2 pieces of plastic is a Poisson random variable. The pinholes occur randomly and independently of one another. The random variable X is defined to be the number of pin- holes per square yard piece and can assume any number between 0 and a very large value. The interval in this example is an area of 1 yd2.

### Poisson Probability Formula

The probability of x occurrences of some event in an interval where the Poisson assumptions of randomness and independence are satisfied is given by formula (5.10), where λ is the mean number of occurrences of the event in the interval and the value of e is approximately 2.71828. The value of e is found on most calculators and powers of e are easily evaluated. Tables of Poisson probabilities are found in many statistical texts. However, with the wide spread availability of calculators and statistical software, they are being less widely utilized.

The mean of a Poisson random variable is given by

μ = λ             (5.11)

The variance of a Poisson random variable is given by

σ2 = λ             (5.12)

EXAMPLE 5.24 The number of small pinholes in sheets of plastic are of concern to a manufacturer. If the number of pinholes is too large, the plastic is unusable. The mean number per square yard is equal to 2.5. The 1-yd2 sheets are unusable if the number of pinholes exceeds 6. The probability of interest is P(X > 6), where X represents the number of pinholes in a 1-yd2 sheet. This probability is found by finding the probability of the complementary event and subtracting it from 1.

P(X > 6) = 1 – P(X ≤ 6)

The solution is given by STATISTIX and is shown in Fig. 5-1. Note that P(X ≤ 6) = 0.98581 is given in the Results portion of the STATISTIX dialog box.

P(X > 6) = 1 – P(X ≤ 6) = 1 - 0.98581 = 0.0142

By multiplying 0.0142 by 100, we see that 1.42% of the plastic sheets are unusable.

EXAMPLE 5.25 The Poisson distribution approximates the binomial distribution closely when n ≥ 20 and p ≤ 0.05. Consider the following MINITAB output created by the pull downs Calc Probability Distributions Binomial and Calc Probability Distributions Poisson.

Cumulative Distribution Function

Binomial with n = 150 and p = 0.01

x           P(X < = x)
2           0.809482

Letting λ = np and figuring the Poisson probability we get the following:

Cumulative Distribution Function

Poisson with mean = 1.5

x           P(X < = x)
2           0.808847

The binomial probability of the event X ≤ 2 is 0.809482 and the Poisson Approximation of the same event is 0.808847.

Practice problems for these concepts can be found at:

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