hw2e - Winter 2003 School of Electrical Engineering Course:...

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Unformatted text preview: Winter 2003 School of Electrical Engineering Course: Probability and Statistics Problem Set #1 1. A. Consider a sequence of independent trials. Each trial can have two outcomes: S (success) or F (failure). In the n-th trial the probability of S is p n , where 0 ≤ p n < 1. Let T be the number of trials needed for one S (i.e. if the 1st trial is S , then T = 1, if the 1st trial is F and the 2nd trial is S , then , T = 2, etc.). If all trials are F , then we say that T = ∞ . Find a simple necessary and sufficient condition on { p n } so that P { T < ∞} = 1 . Hint . If r n = p n / (1- p n ), show that e- r n ≤ 1- p n ≤ e- p n and hence that e- ∑ ∞ n =1 r n ≤ ∞ Y n =1 (1- p n ) ≤ e- ∑ ∞ n =1 p n . B. Find the probability P { T < ∞} for the following cases: (i) p n = p , n ≥ 1, where 0 < p < 1; (ii) p n = 1 /n , n ≥ 2, p 1 = 0; (iii) p n = 1 /n 2 , n ≥ 2, p 1 = 0; (iv) p n = 1 / (4 n 2 ), n ≥ 1. Hint . sin( πz ) = πz Q ∞ n =1 (1- z 2 n 2 )....
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hw2e - Winter 2003 School of Electrical Engineering Course:...

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