Transcript 7_Distributions

f(x)
Uniform Distribution
0.6
0.4
0.2
0
2.5
3.5
4.5
5.5
X
F(x)
Uniform Distribution
1.00
0.75
0.50
0.25
0.00
2.5
3.5
4.5
5.5
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.
Exponential Distribution
2.5
f(X)
2
1.5
1
0.5
0
0
5
10
15
X
1.00
F(X)
0.75
0.50
0.25
0.00
0
5
10
15
X
Lifetime of objects with constant hazard rate
Times 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 lifetimes
Can approximate normal, but is restricted to being
a positive number
Beta Distribution
Very flexible distribution – can approximate
almost anything, but with little theoretical
basis
Kolmogorov-Smirov Test
Expected
1.00
F(X)
0.75
Observed
0.50
0.25
0.00
0
5
10
X
15
Chi-Square Test
0
Successes
1
Success
2
Successes
Observed
12
5
3
Expected
10
5
5
∑{[(O-E)^2]/E}
Bernoulli Trial
Yes
0
No
0.72
Basically a “yes”/”no” outcome
Parameter is p – probability of “yes”
In this example, p=0.72
1
Multinomial
Age 0
0
Age 1Age 2+
0.45
0.66
Multiple categorical outcomes
Parameters are p for each category
1
Binomial Distribution
Number of success in t independent trials
Geometric Distribution
Number of failures before a success
Number 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 events
Note 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