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131A_1_HW3(1) - 90 Chapter 2 Basic Concepts of Probability...

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Unformatted text preview: 90 Chapter 2 Basic Concepts of Probability Theory FlGURE P2.4 Section 2.5: Independence of Events th outcomes are 12 3 4} andA : {1,2},B = {1,3},C : {1.4}.Assuine e 2.82. Lghifoeriible aAi‘e A. B, and C independent events? e . ‘_ . . A:{0<U<t/2}. . the unit Inteival. Let ‘ . (1 nt? 2.83. Let U be $160“;le ring 0:1 {321: U < 1}. Are any of these events indepefl e B={1/4<U< wan ' etball court after school. Alice makes .fi ec 2.84. Alice and Mary practice free ?fi$§a;§1:::;:m with probability pm;:1:1dp:§:ep:::1::lla ihYOWS With Pl‘Opablhthfoaégs when Alice and Mary each take one s o . ity of the. $110233: C(:lrlMary scores a basket; both score; both miss;i BC AC and B‘ and Ac baSkEtiflth: :1 and B are independent events, then the pairs A an . i w t a 1 2.85' 2:: Bc are also independent. . .f HAW] : HA 136]. hat events A and B are independent 1 . d P[C]‘ 2.86. Show tB nd C be events with probabilities P[A], PlB], an 2.87. 1:; [Find :[A U B] if A and B are independent.1 USive ' d PlA U B] if A and B are mutually exc . (b) Fl'nd PlA U B U C 1 if A, B, and C are independent. 1 ( dusk/a (C) Find PEA U B U Cl if A. B, and C are pairWise mutual 3 ex (11) Fin , , . lecting a ball . . us at random and then se . d” . _ . ts of ickmg one of two ur nt “um I is selecte 2'88. An experiment Softiiig its Eolor (black or white). Let A be thigh; are A and B indc. from thii um ant “2 black ball is observed.” Under what con 1 and B t e even . t r’ . , .and C are independfir1 - ganja: probabilities in Problem 2.14 assumlng that events 3.1515,. in Problem 2,15. As— 2.89- 1n S of systems at . - ‘th . ‘ ' ' t the three tyl’e , k unit falls wr 2.90. Find the prtilbabliltlstlie: 31‘: system fail independently and that a type sume that at UD . ~ the Pmbablmy p k ‘ - u u - Problem 2,16. Assume that all units in ' d the robabilities that the system is up 1n. f. .15 with probability Pk- 2.91. Fint m f; independently and that a type k unit a1 ff and the occurrence Of events sys e . . , mber o imes . 9 ‘ . e eatcd a laugh nu . d endent. 2.92. A random exgedlngfwtvbulld you test whether events A and B are 1111:2011 test whether A and B is no e ' ‘ 1 haracters. How wou ‘ . — . ‘e of hexadecuna C . .‘ t With indepen 2.93. ConSider a \f/el'y 12:51::2liih: four bits in the hex characters are consmten the relative requ ‘ 1 n s ‘ f ‘Oin? ~ an )7 when a second («0 (km tosse: brobability of the system in Example 2-35 bemg up a 0/1 (“‘nmniite e Problems 91 2.95. In the binary communication system in Example 226, find the value of a for which the input of the channel is independent of the output of the channel. Can such a channel be used to transmit information? 2.96. In the ternary communication system in Problem 2.81, is there a choice of e for which the input of the channel is independent of the output of the channel? Section 2.6: Sequential Experiments 2.97. A block of 100 bits is transmitted over a binary communication channel with probability of bit error p = 10‘2. (a) If the block has 1 or fewer errors then the receiver ac ability that the block is accepted. (b) If the block has more than 1 error, then the block is retrans ity that M retransmissions are required. cepts the block. Find the prob- mitted. Find the probabil- 2.98. A fraction 1) of items from a certain production line is defective. (a) What is the probability that there is more than one defective item in a batch of n0 items? (b) During normal production p = 10‘3 but when production malfunctions p = 101"]. Find the size of a batch that should be tested so that if any items are found defective we are 99% sure that there is a production malfunction. 2.99. A student needs eight chips of a certain type to build a circuit. It is known that 5% of these chips are defective. How many chips should he bu y for there to be a greater than 90% probability of having enough chips for the circuit? 2.100. Each of n terminals broadc asts a message in a given time slot with probability p. (it) Find the probability that exactly one terminal tr ansmits so the message is received by all terminals without collision. (b) Find the value of p that maximizes the probability of successful transmission in part a. (c) Find the asymptotic value of the probability of successful transmission as‘n becomes large. 2.101. A system contains eight chips. The lifetime of each chip has a Weibull probability law: with parameters A and k = 2: P[(t, 00)] = e‘imk for t 2 0. Find the probability that at least two chips are functioning after 2/)t seconds. A machine makes errors in a certain 0 errors. The fraction of errors that are type 1 is (r, and type 2 is 1 ~ a. (a) What is the probability of k errors in n operations? ([1) What is the probability of k] type 1 errors in n operations? (c) What is the probability of k2 type 2 errors in 11 operations? (d) What is the joint probability of k1 and k2 type 1 and 2 errors, respectively, in n opera— tions? 2.102. peration with probability p. There are two types of 2.103. Three types of packets arrive at a router port. Ten percent of the packets are “expedited forwarding (EF).” 30 percent are “assured forwarding (AF),” and 60 percent are “best efv fort (BE).” (3) Find the probability that k of N packets are not expedited forwarding. (b) Suppose that packets arrive one at a time. Find the probability that k packets are received before an expedited forwarding packet arrives. (c) Find the probability that out of 20 packets, 4 are EF packets, 6 are AF packets. and in ...
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