The Treaty of Versailles
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Finding the maximum likelihood estimator
Equations are omitted for technical reasons - download the original pdf
The experimental outcome was that a flowers were red, b pink and c white. This gives a total of [Equation goes here ; ; download the original to see it.] flowers in the sample. Given that the probability of a red allele is p, we want to find an expression for the probability of the sample containing a flowers were red, b pink and c white and then make this a maximum. If there were just two trials the probability tree would look like this [Diagram goes here ; ; download the original to see it.] The probability of obtaining a pink followed by a red flower would be [Equation goes here ; ; download the original to see it.] But for a pink and a red in any order we have to multiply by the number of branches, which is, in two trials, just . For [Equation goes here ; ; download the original to see it.] trials where a flowers were red, b pink and c white in exactly that order the probability would be [Equation goes here ; ; download the original to see it.] But if we want to find the probability of a red, b pink and c white in any order we must multiply by the total number of branches that give this combination. That is the total number of permutations of a red, b pink and c white flowers. This is a problem in permutations were we have n objects to permute, but there are a identical twins of one kind, b identical twins of another and c identical twins of a third kind. (You may want to look over the unit on permutations and combinations at this point!) The total number of permutations giving a red, b pink and c white flowers is [Equation goes here ; ; download the original to see it.] Thus the probability of a red, b pink and c white flowers in any order is [Equation goes here ; ; download the original to see it.] This gives the probability of the particular experimental outcome as a function of the probability of a red allele, p. We can express this as a function. [Equation goes here ; ; download the original to see it.] This maximum likelihood estimator for the population parameter p.
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Contents of
The Treaty of Versailles
1 The Treaty of Versailles, constraints on the peace-makers
2 Maximum Likelihood Estimators
3 Progress of the Paris Peace Conference and the Treaty of Versailles
4 Finding the maximum likelihood estimator
5 Contemporary reactions to the Treaty of Versailles
6 Determining an estimate for p
7 Assessement of the Treaty of Versailles
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