POISSON, POISSON-GAMMA AND ZERO-INFLATED REGRESSION MODELS OF
CK-12 FlexBook - Poisson and Geometric Probability Distributions
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Poisson & Geometric Probability Distributions Big Picture Study Guides The geometric and Poisson probability distributions are two other discrete probability distributions that are to the binomial distribution. Both of these probability distributions can be applied to solve real-world proble geometric probability distribution is useful for determining the number of trials needed to achieve success, whi Poisson distribution is useful for describing the number of events that will occur during a specific interval of 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 specifi interval or space. Geometric Probability Distribution Experiment • • • • Experiment consists of a sequence of independent trials Each trial has two outcomes: success or failure The number of trials is not fixed; instead, the experiment continues until the first success The probability of success is the same for each trial. Geometric Distribution • • • The geometric distribution is found by calculating the geometric probabilities for n = 1, 2, 3, .... A discrete probability distribution because n can only be whole numbers As n increases, P(x = n) decreases Mean for the geometric distribution: Standard deviation for the binomial distribution: 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? Figure: Geometric probability distributions for three different val- ues of p. Probability • • 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) = (1-p)n-1(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) • Notes ThisguidewascreatedbyLizhiFanandJinYu.Tolearnmoreaboutthestudent authors,visithttp://www.ck12.org/about/about-us/team/interns. Page 1 of 2 v1.1.9.2012
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Ck-12 Flexbook on the Geometric and Poisson Probability explains that they are two other discrete probability distributions that are related to the binomial distribution.
Transcript of CK-12 FlexBook - Poisson and Geometric Probability Distributions
Poisson & Geometric ProbabilityDistributionsBig Picture
Study Guides
The geometric and Poisson probability distributions are two other discrete probability distributions that are relatedto the binomial distribution. Both of these probability distributions can be applied to solve real-world problems. Thegeometric probability distribution is useful for determining the number of trials needed to achieve success, while thePoisson distribution is useful for describing the number of events that will occur during a specific interval of time or ina specific distance, area, or volume.
Key TermsGeometric 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 DistributionExperiment•
••
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Experiment consists of a sequence of independenttrialsEach trial has two outcomes: success or failureThe number of trials is not fixed; instead, theexperiment continues until the first successThe probability of success is the same for eachtrial.
Geometric Distribution
•
•
•
The geometric distribution is found by calculatingthe geometric probabilities for n = 1, 2, 3, ....A discrete probability distribution because n can onlybe whole numbersAs n increases, P(x = n) decreases
Mean for the geometric distribution:
Standard deviation for the binomial distribution:The experiment is essentially binomial trials repeateduntil the first success is achieved, and then theexperiment stops•
•
Example: the number of times a coin needs to betossed until the first head (success) appearsUseful in business applications–example: howmany candidates need to be interviewed beforethe perfect candidate for a job is found?
Figure: Geometric probability distributions for three different val-ues of p.
Probability•
•
Random variable X is the number of trials until thefirst success appearsTo calculate the probability of getting a success onthe nth trial, P(x = n) = (1-p)n-1(p), where n is awhole number and p is the probability of success(this value is the same for each trial)To directly find the probability of more than ntrials completed before there is one success,you would need to sum the probabilities ofan infinite number of trials, which would beimpossible. Instead, use the complement rule:P(x > n) = 1 - P(x ≤ n)
•
Notes
ThisguidewascreatedbyLizhiFanandJinYu.Tolearnmoreaboutthestudentauthors,visithttp://www.ck12.org/about/about-us/team/interns.
Page 1 of 2v1.1.9.2012
Poisson & Geometric Probability
Poisson Probability Distribution Distributions cont.Experiment•
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Experiment consists of counting the number ofevents that will occur during a specific intervalof time or in a specific distance, area, or volumeThere are two outcomes: the event occurs(success) or does not occur (failure)Each event is independentThe probability that an event occurs during thespecified time interval or space is the same
Poisson Distribution
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The Poisson distribution is found by calculating thePoisson probabilities for n = 1, 2, 3, ....A discrete probability distribution because n can only bewhole numbers
Mean for the geometric distribution: μ = λ
Standard deviation for the binomial distribution:
The experiment is a special case where the numberof binomial trials gets larger and the probability ofsuccess gets smaller•
•
Example: the number of traffic accidents at aparticular intersectionUseful in predicting or estimating a number ofthings – planes at an airport, the number offishes caught by a fisherman, arrival times, etc.
Probability•
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Random variable X is the number of events thatoccur (successes)To calculate the probability of n events,
,where λ is the mean number of events in thetime, distance, volume, or area• e is approximately equal to 2.7183
• To directly find the probability of more thanevents occuring, you would need to sum theprobabilities of an infinite number of trials,which would be impossible. Instead, use thecomplement rule.
Figure: Poisson probability distributions for three different values ofλ.
For a binomial distribution where the number of trials n ≥ 100and the probability of success p where np < 100, then thebinomial distribution for k successes can be approximatedwith a Poisson distribution where λ = np
Graphing CalculatorIn a graphing calculator, we can use built-in commands to find the geometric and Poisson distributions.
Geometric DistributionThe command for geometric distribution is: geometpdf(p, x). p is the probability of success, and x is the trial that wewant 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 DistributionThe command for Poisson distribution is: poissonpdf(λ, x). λ is the expected number of events, and x is the number ofevents. 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. Thiswill 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 commandsare found by pressing [2ND][DISTR].
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