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Multivariate Probability Distributions

Multivariate probability distributions are essential in analyzing two or more characteristics observed in experiments. They help study the relationships among variables and predict one based on others. Key types include joint distributions (outcomes of all variable combinations), marginal distributions (outcomes for a single variable), and conditional distributions (outcomes for one variable given others). Additionally, the multinomial distribution extends the binomial distribution for trials with multiple categories, allowing analysis of complex scenarios such as daily defect rates and customer arrivals.

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Multivariate Probability Distributions

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  1. Multivariate Probability Distributions

  2. 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)

  3. Joint Distribution

  4. Marginal Distributions

  5. Conditional Distributions • Describes the behavior of one variable, given level(s) of other variable(s)

  6. Expectations

  7. Expectations of Linear Functions

  8. Variances of Linear Functions

  9. Covariance of Two Linear Functions

  10. 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 • p1+…+pk = 1 • Y1+…+Yk = n

  11. Multinomial Distribution

  12. Multinomial Distribution

  13. Conditional Expectations When E[Y1|y2] is a function of y2, function is called the regression of Y1 on Y2

  14. Unconditional and Conditional Mean

  15. Unconditional and Conditional Variance

  16. 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) Yi|Ai ~ Bin(Ai,p)

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