sol2 - Stat 219 Stochastic Processes Homework Set 2 Fall...

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Unformatted text preview: Stat 219 - Stochastic Processes Homework Set 2, Fall 2008, Due: October 8 See Victor Hu for questions on grading 1. Exercise 1.2.22. Write (Ω , F ,P ) for a random experiment whose outcome is a recording of the results of n independent rolls of a balanced six-sided dice(including their order). Compute the expectation of the random variable D ( ω ) which counts the number of different faces of the dice recorded in these n rolls. ANS: Ω = { 1 , 2 ,..., 6 } n , F = 2 Ω and P ( A ) = | A | 6- n for A ∈ F , where | A | denotes the number of elements in the set A . Now, let A i denote the event that the i-th face has been recorded and I i ( ω ) = I A i the corresponding indicator random variable. Since D ( ω ) = ∑ 6 i =1 I i ( ω ), we have by linearity of the expectation that: E ( D ) = 6 X i =1 E I i = 6 X i =1 P ( A i ) = 6 X i =1 | A | 6- n . The complement of A 1 consists of all sequences ω ∈ Ω whose components are from { 2 , 3 , 4 , 5 , 6 } , so it has 5 n elements. Consequently, | A 1 | = 6 n- 5 n . The same size calculation applies for each A i , leading to E ( D ) = 6(1- (5 / 6) n ). 2. Exercise 1.2.27. Show that a R.V. X is integrable if and only if E | X | I | X | >M → 0 as M →∞ . ANS: ( ⇐ ) Since lim M E | X | I | X | >M = 0, ∃ M and > 0 s.t. E | X | = Z | X |≤ M | X | dP + Z | X | >M | X | dP ≤ M P ( {| X | ≤ M } ) + < ∞ since P is a probability measure. ANS: ( ⇒ ) The easiest way to show this is to use monotone convergence or dominated convergence. (Monotone convergence) As M → ∞ , | X | I | X |≤ M ↑ | X | and so by the monotone convergence theorem E I | X |≤ M ↑ E | X | . Also, since E | X | = E I | X |≤ M + E I | X | >M taking limits we have E | X | = E | X | + lim M →∞ E I | X | >M . But since E | X | < ∞ we must have lim M →∞ E I | X | >M = 0. (Dominated convergence) | X | I | X | >M → 0 a.s. as M → ∞ . Then since | X | I | X | >M ≤ | X | and E | X | < ∞ we have lim M →∞ E | X | I | X | >M = 0 by the dominated convergence theorem....
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This note was uploaded on 11/08/2009 for the course STAT 219 taught by Professor -2 during the Fall '08 term at Stanford.

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sol2 - Stat 219 Stochastic Processes Homework Set 2 Fall...

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