starting point for generating other distributions
DESCRIPTION
Starting point for generating other distributions. Normal Distribution. Commonly used – processes where many random variables are added results in normal distribution. Lognormal Distribution. Perhaps not as commonly recognized or used as the - PowerPoint PPT PresentationTRANSCRIPT
Uniform Distribution
00.20.40.6
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f(x)
Uniform Distribution
0.000.250.500.751.00
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F(x
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Starting point for generating other distributions
Normal Distribution
Commonly used – processes where many random variables are added results in normal distribution
Lognormal Distribution
Perhaps not as commonly recognized or used as the normal distribution, but often more appropriate. Processes where many random variables are multiplied results in lognormal distribution. Note that most differential equations result from sequential multiplication of rates, so this is often the result.
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f(X
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F(X
)Exponential Distribution
Lifetime of objects with constant hazard rateTimes between independent events (waiting time)
Gamma and Erlang Distribution
Time to complete task when have several independent steps (waiting time)Gamma – more general, Erlang restricted to alpha as a positive integer
Weibul Distribution
Also used to generate device lifetimesCan approximate normal, but is restricted to beinga positive number
Beta Distribution
Very flexible distribution – can approximatealmost anything, but with little theoretical basis
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)Kolmogorov-Smirov Test
Expected
Observed
Chi-Square Test
0 Successes
1 Success
2 Successes
Observed 12 5 3
Expected 10 5 5
∑{[(O-E)^2]/E}
Bernoulli Trial
Basically a “yes”/”no” outcomeParameter is p – probability of “yes”In this example, p=0.72
0 10.72
Yes No
Multinomial
Multiple categorical outcomesParameters are p for each category
0 10.45
Age 0 Age 1Age 2+
0.66
Binomial Distribution
Number of success in t independent trials
Geometric Distribution
Number of failures before a successNumber of items examined before a defect found
Negative Binomial Distribution
Often describes number of animals in a quadrat, particularly when animals are clustered, as might happen for schooling animals, or animals with patchy habitats
Poisson Distribution
Occurrence of rare eventsNote that the variance=mean for this distribution
Generating Random Observations
Based on Transformation of U(0,1)
•Inversion of distribution function•Special relationship between distributions e.g., convolution•Acceptance-rejection methods
Transformation of U(0,1) to get exponential
Box-Mueller method for generating normal
Exponentiate normal to get lognormal
Erlang – sum of m exponential distributions
Rejection Method