Gabarito-Prova1-CE069-2009.pdf - UNIVERSIDADE FEDERAL DO...

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UNIVERSIDADE FEDERAL DO PARANÁ SETOR DE CIÊNCIAS EXATAS DEPARTAMENTO DE ESTATÍSTICA GABARITO PRIMEIRA PROVA DE PROBABILIDADE B (CE069) Prof. Benito Olivares Aguilera 1 o Sem./09 1) a) Suponha que X e Y são variáveis aleatórias independentes e identicamente distribuídas, assumindo os valores -1 e 1 com a mesma probabilidade. Podem as variáveis X e Z=XY serem independentes? Verifique. R: Do enunciado temos que X e Y são discretas, com distribuição 2 1 ) 1 ( ) 1 ( e 2 1 ) 1 ( ) 1 ( = = = - = = = = - = Y P Y P X P X P . A variável Z=XY, por ser produto, assume também valores -1 e 1 com probabilidades 2 / 1 2 / 1 2 / 1 2 / 1 2 / 1 ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 , 1 ( ) 1 , 1 ( ) 1 ( ) 1 ( = × + × = - = = + = - = = - = = = - = = - = = - = Y P X P Y P x P Y X P Y x P XY P Z P Analogamente, 2 / 1 ) 1 ( = = Z P . Determinemos a distribuição conjunta de X e Z : 4 / 1 ) 1 , 1 ( ) 1 , 1 ( ) 1 , 1 ( = = - = = - = - = = - = - = Y X P XY X P Z X P , 4 / 1 ) 1 , 1 ( ) 1 , 1 ( ) 1 , 1 ( = - = - = = = - = = = - = Y X P XY X P Z X P . Os outros valores calculam-se da mesma forma e aparecem na tabela a seguir: X Z -1 1 Marginal de Z -1 1/4 1/4 1/2 1 1/4 1/4 1/2 Marginal de X 1/2 1/2 1 Notamos que } 1 , 1 { , ), ( ) ( ) , ( - 2200 = = = = = z x z Z P x X P z Z x X P . Portanto X e Z=XY são independentes.
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b) Sejam X e Y variáveis aleatórias com funções de distribuição X F e Y F , respectivamente. Poderíamos definir uma distribuição conjunta como sendo ) ( ) ( ) , ( , y F x F y x F Y X Y X + = ? Explique.
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