Multivariate Probability Distributions

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Multivariate Probability Distributions
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Multivariate Random Variables

• In many settings, we are interested in 2 or more
  characteristics observed in experiments
• Often used to study the relationship among
  characteristics and the prediction of one based
  on the other(s)
• Three types of distributions
• Joint Distribution of outcomes across all
  combinations of variables levels
• Marginal Distribution of outcomes for a single
  variable
• Conditional Distribution of outcomes for a
  single variable, given the level(s) of the other
  variable(s)

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Joint Distribution
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Marginal Distributions
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Conditional Distributions

• Describes the behavior of one variable, given
  level(s) of other variable(s)

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Expectations
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Expectations of Linear Functions
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Variances of Linear Functions
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Covariance of Two Linear Functions
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Multinomial Distribution

• Extension of Binomial Distribution to experiments
  where each trial can end in exactly one of k
  categories
• n independent trials
• Probability a trial results in category i is pi
• Yi is the number of trials resulting in category
  I
• p1pk 1
• Y1Yk n

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Multinomial Distribution
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Multinomial Distribution
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Conditional Expectations
When EY1y2 is a function of y2, function is
called the regression of Y1 on Y2
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Unconditional and Conditional Mean
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Unconditional and Conditional Variance
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Compounding

• Some situations in theory and in practice have a
  model where a parameter is a random variable
• Defect Rate (P) varies from day to day, and we
  count the number of sampled defectives each day
  (Y)
• Pi Beta(a,b) Yi Pi Bin(n,Pi)
• Numbers of customers arriving at store (A)
  varies from day to day, and we may measure the
  total sales (Y) each day
• Ai Poisson(l) YiAi Bin(Ai,p)