Practice Questions 6 Answer

Practice Questions 6 Answer - ANSWERS TO PRACTICE QUESTIONS...

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Unformatted text preview: ANSWERS TO PRACTICE QUESTIONS 6 – ACTSC 431/831, FALL 2011 1. (a) The probability is Pr { N 3 12 = 1 ,N 5 12 = 2 } = Pr { N 3 12 = 1 } Pr { N 2 12 = 1 } = 0 . 02747 . (b) The probability is Pr { N 1 12 = 0 | N 1 2 = 3 } = Pr { N 1 12 =0 } Pr { N 5 12 =3 } Pr { N 1 2 =3 } = 0 . 5787 . (c) The probability is 5 2 ! (Pr { X i > 1 } ) 2 (Pr { X i ≤ 1 } ) 3 = 0 . 34183 . (d) The probability is Pr { N 1 = 2 ,X 1 > 1 ,X 2 > 1 } = Pr { N 1 = 2 } Pr { X 1 > 1 } Pr { X 2 > 1 } = 0 . 02489 . (e) The expected time of the 2nd claim is E T 2 | T 6 = 20 12 = E W 1 + W 2 | W 1 + ··· + W 6 = 20 12 = 2 E W 1 | W 1 + ··· + W 6 = 20 12 = 2 × 20 12 × 1 6 = 5 9 . (f) Let ψ n ( u ) be the probability that ruin occurs on or before the n th claim, n = 1 , 2 .... For any u ≥ 0, ψ 1 ( u ) = R ∞ e- ( u +1 . 2 t ) e- t dt = e- u 2 . 2 . Furthermore, for any u ≥ 0, ψ 2 ( u ) = Z ∞ " Z u +1 . 2 t e- ( u +1 . 2 t- x ) 2 . 2 e- x dx + e- ( u +1 . 2 t ) # e- t dt = 1 . 2 2 . 2 3 + 1 2 . 2 + u 2 . 2...
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This note was uploaded on 02/08/2012 for the course ACTSC 431 taught by Professor Laundriualt during the Fall '09 term at Waterloo.

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