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Tom's edit of one lecture July 24
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lectures/prob_matrix.md

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The joint distribution induce marginal distributions
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$$
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\textrm{Prob}\{X=i\}= \sum_{j=0}^{J-1}f_{ij} = \mu_i, i=0,\ldots,I-1,
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\textrm{Prob}\{X=i\}= \sum_{j=0}^{J-1}f_{ij} = \mu_i, \quad i=0,\ldots,I-1
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$$
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$$
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\textrm{Prob}\{Y=j\}= \sum_{i=0}^{I-1}f_{ij} = \nu_i, i=0,\ldots,J-1
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\textrm{Prob}\{Y=j\}= \sum_{i=0}^{I-1}f_{ij} = \nu_j, \quad j=0,\ldots,J-1
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$$
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For example, let the joint distribution over $(X,Y)$ be

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