## 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: