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

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 )....
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online