hw3 - Homework 3 On-campus students: Due by 10:40 AM on...

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Homework 3 On-campus students: Due by 10:40 AM on Monday, Oct. 3rd, 2011 Off-campus students: Due by 11:55 PM on Monday, Oct. 10th, 2011 (As usual, the problems labeled “SS” are for self-study and should not be submitted for grading.) B INOMIAL , G EOMETRIC , AND P OISSON P ROBABILITIES SS-1. On a very noisy communication channel, the probability of a packet being received correctly is 0.2. If the packet is not received correctly, then the destination does not send an acknowledgment (ACK). The source repeatedly transmits the packet until an ACK is received. (a) What is the probability that three transmissions are required? [0.128] (b) What is the probability that more than three transmissions are required? [0.512] (c) Given that the first three transmissions were failures, what is the probability that more than three more transmissions are required? [0.512] (d) Find a formula for the probability that Δ more transmissions are required given that the first j transmissions were failures. [ ( 1 - p ) Δ - 1 p ] (e) Explain why geometric probabilities are said to be “memoryless”. 1. In 2003, there were many media reports about the number of shark attacks in Florida. At the end of the year, there were a total of 30 unprovoked shark attacks. By comparison, there were 246 shark attacks over the prior ten years. (a) Give an expression for the probability of more than thirty shark attacks occurring in a year based on the historical data (don’t include the data for 2003). Use a Gaussian approximation to give an approximate numerical answer. (b) Suppose that the probability that there are more than 30 shark attacks in a given year is 0.1. Find the probability that there is at least one year with more than 30 shark attacks in 10 year period. (c) Use the historical data to answer this question. Shark attacks occur primarily during warm weather, so suppose that shark attacks only occur over a 30 week period. In 2003, the media made a big deal about two shark attacks during the same week. What is the probability of having at least two shark attacks during a given week? (d) Using the assumptions of the previous part of this problem, what is the probability of having at least one week with three or more shark attacks during a given year?
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2. (Modified from Stark and Woods , 3rd ed.) (Independence of events in disjoint arrivals for Poisson law) The average number of cars arriving at a tollbooth per minute is λ and the probability of k cars in the interval ( 0 , T ) is P ( k ;0 , T ) = e - λ T [ λ T ] k k ! . Consider two disjoint, that is nonoverlapping intervals, say ( 0 , t 1 ) and ( t 1 , T ) . Then for the Poisson law: P [ n 1 cars in ( 0 , t 1 ) and n 2 cars in ( t 1 , T )] = P [ n 1 cars in ( 0 , t 1 )] P [ n 2 cars in ( t 1 , T )] Thus events in disjoint intervals are independent. (a) Show that
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hw3 - Homework 3 On-campus students: Due by 10:40 AM on...

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