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Relationship between Ploidy, Alleles and Genotypes (view source)

Revision as of 10:27, 2 February 2015

724 bytes removed ,  10:27, 2 February 2015
→‎Derivation for counting the number of genotypes
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= Derivation for counting the number of genotypes =
 
= Derivation for counting the number of genotypes =
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For the case where A < P,  there must always be P observed alleles and there can only be at most A alleles.  This can be modeled by P+A-1 points where you choose A-1 points to be dividers to define the alleles.  
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There must always be P observed alleles and there can only be at most A alleles.  This can be modeled by P+A-1 points where you choose A-1 points to be dividers that separate the alleles.  
 
Thus the number of ways you can observe this is <math> \binom{P+A-1}{A-1}  </math>.
 
Thus the number of ways you can observe this is <math> \binom{P+A-1}{A-1}  </math>.
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For the case where A >= P,
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<math>
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  \begin{align}
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F(P,A) &=  \sum_{i=1}^{P} \| \{\text{i distinct alleles observed in a set of P alleles}\} \| \\
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          &=  \sum_{i=1}^{P-1} \| \{\text{i distinct alleles observed in a set of P alleles}\} \|  + \| \{\text{P distinct alleles observed in a set of P alleles}\} \| \\
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          &=  \sum_{i=1}^{P-1} \| \{\text{i distinct alleles observed in a set of P alleles}\} \|  + \binom{A}{P}\\
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          &=  \sum_{i=1}^{P-1} \| \{\text{ways to choose i distinct alleles}\} \|\| \{\text{ways to arrange i distinct alleles in a set of P alleles}\} \|  +\binom{A}{P} \\
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&=  \sum_{i=1}^{P-1} \binom{A}{i}\binom{P-1}{i-1}  + \binom{A}{P}
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  \end{align}
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</math>
      
= Derivation for getting the index of a genotype  in an enumerated list =
 
= Derivation for getting the index of a genotype  in an enumerated list =
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