HW4_Sol - 5 - ECEN 646 Homework 4 Solutions 1 C ¸inlar...

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ECEN 646 Homework 4 Solutions 1 C ¸inlar Exercise (5.11), Chapter 3 The rate of crossings is 4 vehicles per minute. So, the probability of a vehicle passing a point in a given second is 4 60 = 0 . 067. (a) p = P [ X n = 1] = P [a vehicle passing in a given second] = 1 15 = 0 . 067 (b) P [ T 4 - T 3 = 12] = P [It takes 12 trials after the 3rd success for the 4th success to occur] = P [3rd success is followed by 11 failures and a success] = q 11 p = 0 . 0312 (c) E [ T 4 - T 3 ] = E [time between the 3rd and the 4th successes] = E [time for the 1st success] = 1 p = 15 E [ T 13 - T 3 ] = E [time between the 3rd and the 13th successes] = E [time for the first 10 successes] = 10 p = 150 (d) Var ( T 2 + 5 T 3 ) = Var (6 T 2 + 5( T 3 - T 2 )) = Var (6 T 2 ) + Var (5( T 3 - T 2 )) This follows from the independent increments property that makes T 2 and T 3 - T 2 independent. We then use the fact that the variance of a sum of independent r.v’s equals the sum of the individual variances. Var (6 T 2 ) = 36Var ( T 2 ) = 36 2 q p 2 = 15120 Var (5( T 3 - T 2 )) = 25Var ( T 3 - T 2 ) = 25 q p 2 = 5250 Therefore Var ( T 2 + 5 T 3 ) = 20370. 1 This study resource was shared via CourseHero.com
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2 C ¸inlar Exercise (5.12), Chapter 3 Let f ( a 0 , a 1 , . . . a 7 ) be defined as P [ T 8 = 17 | T 0 = a 0 , T 1 = a 1 . . . T 7 = a 7 ]. For the 8 th success to happen at the 17 th trial, we need the 7 th success to have happened earlier. So, the conditional probability above will be non- zero only if the event T 7 16 occurs. Assuming this event occurs, we have f ( a 0 , . . . a 7 ) = P [ X a 7 +1 = 0 , . . . , X 16 = 0 , X 17 = 1] = pq 16 a 7 Therefore P [ T 8 = 17 | T 0 , T 1 . . . T 7 ] = f ( T 0 , T 1 , . . . T 7 ) = pq 16 T 7 if the event T 7 16 occurs. 3 C ¸inlar Exercise (5.18), Chapter 3 (a) For the duel to last exactly 13 rounds, both duelists should stay alive for the first 12 rounds and at least one of them should die in the 13 th round. P [Both duelists survive a round] = P [both shots miss the mark] = (1 - p 1 )(1 - p 2 ) P [At least one person dies in a round] = 1 - P [no one dies in the round] = 1 - (1 - p 1 )(1 - p 2 ) = p 1 + p 2 - p 1 p 2 . Therefore, P [Duel lasts exactly 13 rounds] = (1 - p 1 ) 12 (1 - p 2 ) 12 ( p 1 + p 2 - p 1 p 2 ) (b) The young man comes out alive if he shoots down the opponent first. Consider the possibility that the young man emerges the winner in the n th round. For this to happen, the first ( n -
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