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# chapter5 - 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)
Joint Distribution ( 29 ( 29 ( 29 Variables Random of number any to s Generalize ) , ( , ) , ( 1 ) , ( 0 ) , ( : Function) Density ty (Probabili Case Continuous ) , ( , ) , ( 1 ) , ( 0 , ) , ( : Function) Mass ty (Probabili Case Discrete 1 2 1 1 2 2 1 2 1 2 2 1 2 2 1 1 2 1 1 2 2 1 2 1 2 1 2 2 1 1 2 1 all all 2 1 2 2 1 1 2 1 ∫ ∫ ∫ ∫ ∑ ∑ ∑ ∑ - - - - -∞ = -∞ = = = = = = = = = = y y y t y t y y dt dt t t f y Y y Y P y y F dy dy y y f y y f y y p y Y y Y P y y F y y p y Y y Y P y y p

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Marginal Distributions ables) other vari all over integrate or (sum Variables Random of number any to s Generalize ) , ( ) ( ) , ( ) ( : Case Continuous ) , ( ) ( ) , ( ) ( : Case Discrete 1 2 1 2 2 2 2 1 1 1 1 all 2 1 2 2 all 2 1 1 1 2 - - = = = = dy y y f y f dy y y f y f y y p y p y y p y p y y
Conditional Distributions Describes the behavior of one variable, given level(s) of other variable(s) ( 29 ( 29 1 2 1 2 1 1 2 1 1 2 2 1 2 1 2 2 2 1 2 1 1 all 1 2 1 1 2 1 1 1 2 2 1 2 2 all 2 1 2 2 2 1 2 2 1 1 2 1 1 ) | ( ) ( ) , ( ) | ( 1 ) | ( ) ( ) , ( ) | ( : Case Continuous 1 ) | ( ) ( ) , ( | ) | ( 1 ) | ( ) ( ) , ( | ) | ( : Case Discrete 2 1 y dy y y f y f y y f y y f y dy y y f y f y y f y y f y y y p y p y y p y Y y Y P y y p y y y p y p y y p y Y y Y P y y p y y 2200 = = 2200 = = 2200 = = = = = 2200 = = = = = -

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