Uniform Distribution

00.20.40.6

2.5 3.5 4.5 5.5

X

f(x)

Uniform Distribution

0.000.250.500.751.00

2.5 3.5 4.5 5.5X

F(x

)

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.

0

0.5

1

1.5

2

2.5

0 5 10 15

X

f(X

)

0.00

0.25

0.50

0.75

1.00

0 5 10 15

X

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

0.00

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0 5 10 15

X

F(X

)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