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Comment: Fixed broken link



 

 

The size of D and M are fundamental properties of the hypergeometric stochastic system like p for a binomial process and l for a Poisson process. Distributions of our uncertainty about the value of these parameters can be determined from Bayesian inference, given a certain sample size taken from the population M, of which s belonged to the sub-population D. The hypergeometric probability of s successes in n samples from a population M of which D have the characteristic of interest is given as:

...


LaTeX Math Inline
bodyp(s)=\frac{\left( \begin{array}{c} D \\ s \end{array} \right) \left( \begin{array}{c} M-D \\ n-s \end{array} \right)}{\left( \begin{array}{c} M \\ n \end{array} \right) }

LaTeX Math Inline
body0\leq s\leq n
,

LaTeX Math Inline
bodys\leq D
,

LaTeX Math Inline
bodyn \leq M

...


So, with a Uniform prior, we get the following posterior densities for D and M:

 

...



LaTeX Math Block
alignmentleft
p(D)=\frac{\left(
    \begin{array}{c}
      D \\
     s
    \end{array}
	 \right)
\left(
    \begin{array}{c}
      M-D \\
     n-s
    \end{array}
	 \right)}{\left(
    \begin{array}{c}
      M \\
     n
    \end{array}
	 \right) }
	 \propto
\frac{D!(M-D)!}{(D-s)!(M-D-n+s)!}

...

LaTeX Math Block
alignmentleft
p(M)=\frac{\left(
    \begin{array}{c}
      D \\
     s
    \end{array}
	 \right)
\left(
    \begin{array}{c}
      M-D \\
     n-s
    \end{array}
	 \right)}{\left(
    \begin{array}{c}
      M \\
     n
    \end{array}
	 \right) }
	 \propto
\frac{(M-D)!(M-n)!}{(M-D-n+s)! M!}

...


These formulae do not equate to standard distributions, and need to be normalized, which can be done easily with Crystal Ball and Excel as shown in the Excel example in the previous topic.

 

 

 

 

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