provaF.pdf - PROVA FINAL TEORIA DAS PROBABILIDADES 2017...

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PROVA FINAL TEORIA DAS PROBABILIDADES - 2017 Quest˜ ao 1. (2,5 Pontos) Seja Ω = R e F := { A R : A ´ e enumer´avel ou A c ´ e enumer´avel } (lembre-se que o vazio ´ e enummeravel). Defina P sobre elementos de F da seguinte forma: P ( A ) := 0 se A for enumer´avel, e P ( A ) := 1 se A c for enumer´avel. Mostre que , F , P ) define um espa¸ co de probabilidade, i.e. que F ´ e uma σ -´algebra e P ´ e uma probablidade. Mostre que se Y 0 ´ e uma vari´avel aleat´oria sobre , F , P ) e p (0 , ) ent˜ao E ( Y p ) = Z 0 py p - 1 P ( Y > y ) dy . Quest˜ ao 2. (2,5 pontos) Sejam X e Y vari´aveis aleat´orias independentes e ambas com distribui¸ c˜ao normal padr˜ao. Determine a distribui¸ c˜ao condicional de ( X, Y ) dado X 2 + Y 2 . Sejam X 1 , X 2 , . . . vari´aveis aleat´orias i.i.d. com distribui¸ ao exponencial de parˆ ametro λ > 0 , e N uma vair´avel independente de ( X n ) 1 com distribui¸ c˜ao geom´ etrica de parˆametro q (0 , 1) . Calcule a m´ edia e a variˆancia de Z := N j =1 X j . Quest˜ ao 3. (2,5 pontos) Sejam X 1 , X 2 , . . . vari´aveis aleat´orias i.i.d. com densidade f ( x ) = 1 π (1 + x 2 ) , x R . Mostre que para todo L > 0 , P ( | X n | > Ln infinitas vezes ) = 1 . Considere a mesma sequˆ encia X 1 , X 2 . . . do item anterior. Determine se ¯ X n := 1 n X k =1 X k , converge quase certamente ou em distribui¸ ao. E no caso afirmativo, determine o limite.
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  • Spring '15
  • vivianlew

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