Random distribution

Random spatial distribution is simulated using poisson distribution.

Simplest example: 100 people are fishing in the same lake for the same time (e.g. 3 h); they have equal probability to catch a fish per unit time. Question: How many fishers catch 0, 1, 2, 3 etc. fish?

No. of fish captured, i

No. of fishers n(i)

Proportion of fishers p(i)

Poisson distribution n'(i)=Np'(i)

0

11

0.11

10

1

25

0.25

23

2

21

0.21

27

3

25

0.25

20

4

9

0.09

12

5

7

0.07

5

6

2

0.02

2

7

0

0.00

1

Total

N=100

1.00

100

Mean number of fish captured by 1 fisher, M = 2.30, and standard deviation, SD = 1.41. Poisson distribution is described by equation:

where m is the mean and i!= 1x2x3x ... xi, 0!=1; 1!=1. Theorem: In poisson distribution, mean = variance: m

Two main methods of parameter estimation

2. Non-linear regression (iterative approximation)

In the table above we used the method of moments: m = M = 2.3.

Chi-square test is used to test if sample distribution is different from theoretical distribution. The equation is:

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