Poisson & Geometric Probability Distributions Big Picture
The geometric and Poisson probability distributions are two other discrete probability distributions that are related to the binomial distribution. Both of these probability distributions can be applied to solve realworld problems. The geometric probability distribution is useful for determining the number of trials needed to achieve success, while the Poisson distribution is useful for describing the number of events that will occur during a specific interval of time or in a specific distance, area, or volume.
Key Terms Geometric Distribution: The discrete probability distribution of the number of trials needed to achieve success. Poisson Distribution: The discrete probability distribution of the number of events that occur in a specific time interval or space.
Geometric Probability Distribution Experiment
Experiment consists of a sequence of independent trials
The geometric distribution is found by calculating the geometric probabilities for n = 1, 2, 3, ....
Each trial has two outcomes: success or failure
The number of trials is not fixed; instead, the experiment continues until the first success
A discrete probability distribution because n can only be whole numbers
As n increases, P(x = n) decreases
The probability of success is the same for each trial.
The experiment is essentially binomial trials repeated until the first success is achieved, and then the experiment stops
Example: the number of times a coin needs to be tossed until the first head (success) appears
Useful in business applications–example: how many candidates need to be interviewed before the perfect candidate for a job is found?
Mean for the geometric distribution: Standard deviation for the binomial distribution:
Figure: Geometric probability distributions for three different val ues of p.
Random variable X is the number of trials until the first success appears
To calculate the probability of getting a success on the nth trial, P(x = n) = (1p)n1(p), where n is a whole number and p is the probability of success (this value is the same for each trial)
To directly find the probability of more than n trials completed before there is one success, you would need to sum the probabilities of an infinite number of trials, which would be impossible. Instead, use the complement rule: P(x > n) = 1 P(x ≤ n)
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Poisson & Geometric Probability Poisson Probability Distribution
Distributions cont . Poisson Distribution
Experiment consists of counting the number of events that will occur during a specific interval of time or in a specific distance, area, or volume
The Poisson distribution is found by calculating the
A discrete probability distribution because n can only be
Poisson probabilities for n = 1, 2, 3, ....
There are two outcomes: the event occurs (success) or does not occur (failure)
Each event is independent
Mean for the geometric distribution: μ = λ
The probability that an event occurs during the specified time interval or space is the same
Standard deviation for the binomial distribution:
The experiment is a special case where the number of binomial trials gets larger and the probability of success gets smaller
Example: the number of traffic accidents at a particular intersection
Useful in predicting or estimating a number of things – planes at an airport, the number of fishes caught by a fisherman, arrival times, etc.
Random variable X is the number of events that occur (successes)
To calculate the probability of n events, , where λ is the mean number of events in the time, distance, volume, or area
• e is approximately equal to 2.7183 •
To directly find the probability of more than events occuring, you would need to sum the probabilities of an infinite number of trials, which would be impossible. Instead, use the complement rule.
Figure: Poisson probability distributions for three different values of λ.
For a binomial distribution where the number of trials n ≥ 100 and the probability of success p where np < 100, then the binomial distribution for k successes can be approximated with a Poisson distribution where λ = np
Graphing Calculator In a graphing calculator, we can use builtin commands to find the geometric and Poisson distributions.
Geometric Distribution The command for geometric distribution is: geometpdf(p, x). p is the probability of success, and x is the trial that we want the success to occur in. This will give us the probability of success occurring on that trial. There is another similar equation called geometcdf, which requires us to plug in two values for x: one low and one high. It will give us the probability of success occurring between those two trials.
Poisson Distribution The command for Poisson distribution is: poissonpdf(λ, x). λ is the expected number of events, and x is the number of events. This will give us the probability that x many events occurred. There is a similar command called poissoncdf, which requires us to plug in two values for x: one low and one high. This will give us the probability the number of events that occurred fell between these two numbers. If you can’t find these commands, check the manual for your graphing calculator. For the TI-83/TI-84, both commands are found by pressing [2ND][DISTR].
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