Starting point for generating other distributions

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Starting point for generating other distributions
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Normal Distribution
Commonly used processes where many random
variables are added results in normal distribution
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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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Exponential Distribution
Lifetime of objects with constant hazard
rate Times between independent events (waiting
time)
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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
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Weibul Distribution
Also used to generate device lifetimes Can
approximate normal, but is restricted to being a
positive number
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Beta Distribution
Very flexible distribution can
approximate almost anything, but with little
theoretical basis
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Kolmogorov-Smirov Test
Expected
Observed
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Chi-Square Test
?(O-E)2/E
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Bernoulli Trial
Yes
No
0
1
0.72
Basically a yes/no outcome Parameter is p
probability of yes In this example, p0.72
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Multinomial
Age 0
Age 1
Age 2
0
1
0.45
0.66
Multiple categorical outcomes Parameters are p
for each category
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Binomial Distribution
Number of success in t independent trials
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Geometric Distribution
Number of failures before a success Number of
items examined before a defect found
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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
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Poisson Distribution
Occurrence of rare events Note that the
variancemean for this distribution
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• Generating Random Observations
• Based on Transformation of U(0,1)
• Inversion of distribution function
• Special relationship between distributions e.g.,
  convolution
• Acceptance-rejection methods

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Transformation of U(0,1) to get exponential
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Box-Mueller method for generating
normal Exponentiate normal to get
lognormal Erlang sum of m exponential
distributions
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Rejection Method