Multivariate Probability Distributions Multivariate Random Variables In many

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Multivariate Probability Distributions

Multivariate Probability Distributions

Multivariate Random Variables • In many settings, we are interested in 2 or more

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)

Joint Distribution

Joint Distribution

Marginal Distributions

Marginal Distributions

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

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

Expectations

Expectations

Expectations of Linear Functions

Expectations of Linear Functions

Variances of Linear Functions

Variances of Linear Functions

Covariance of Two Linear Functions

Covariance of Two Linear Functions

Multinomial Distribution • Extension of Binomial Distribution to experiments where each trial can end

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 • p 1+…+pk = 1 • Y 1+…+Yk = n

Multinomial Distribution

Multinomial Distribution

Multinomial Distribution

Multinomial Distribution

Conditional Expectations When E[Y 1|y 2] is a function of y 2, function is

Conditional Expectations When E[Y 1|y 2] is a function of y 2, function is called the regression of Y 1 on Y 2

Unconditional and Conditional Mean

Unconditional and Conditional Mean

Unconditional and Conditional Variance

Unconditional and Conditional Variance

Compounding • Some situations in theory and in practice have a model where a

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)