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# Starting point for generating other distributions

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1. Starting point for generating other distributions

2. Normal Distribution Commonly used – processes where many random variables are added results in normal distribution

3. 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.

4. Exponential Distribution Lifetime of objects with constant hazard rate Times between independent events (waiting time)

5. 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

6. Weibul Distribution Also used to generate device lifetimes Can approximate normal, but is restricted to being a positive number

7. Beta Distribution Very flexible distribution – can approximate almost anything, but with little theoretical basis

8. Kolmogorov-Smirov Test Expected Observed

9. Chi-Square Test ∑{[(O-E)^2]/E}

10. Bernoulli Trial Yes No 0 0.72 1 Basically a “yes”/”no” outcome Parameter is p – probability of “yes” In this example, p=0.72

11. Multinomial Age 0 Age 1 Age 2+ 0 0.45 0.66 1 Multiple categorical outcomes Parameters are p for each category

12. Binomial Distribution Number of success in t independent trials

13. Geometric Distribution Number of failures before a success Number of items examined before a defect found

14. 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

15. Poisson Distribution Occurrence of rare events Note that the variance=mean for this distribution

16. Generating Random Observations • Based on Transformation of U(0,1) • Inversion of distribution function • Special relationship between distributions e.g., convolution • Acceptance-rejection methods

17. Transformation of U(0,1) to get exponential

18. Box-Mueller method for generating normal Exponentiate normal to get lognormal Erlang – sum of m exponential distributions

19. Rejection Method