Introduction to Probability - Chapter 11 ppsx
Introduction to Probability - Chapter 11 ppsx
... Example 11. 7 is an absorbing Markov chain.3 Which of the genetics examples (Examples 11. 9, 11. 10, and 11. 11) are ab-sorbing?4 Find the fundamental matrix N for Example 11. 10.5 For Example 11. 11, ... . 436 CHAPTER 11. MARKOV CHAINSThus, the common row of W is the unique vector w which is both a fixed rowvector for P and a probability vector. Theorem 11. 8 shows that any fixed row vectorfor ... Experiments,” Annals of Probability, vol. 8 (1980), pp. 117 1 117 6. 11. 2. ABSORBING MARKOV CHAINS 42517 Show that in both Example 11. 11 and the example just given, the probability of absorption...
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Introduction to Probability - Chapter 1 pps
... more appropriate to assign a distributionfunction which assigns probability .513 to the outcome boy and probability .487 to the outcome girl than to assign probability 1/2 to each outcome. This ... easier to compute the probability that an eventdoes not happen rather than the probability that it does. We then use Property 5 to obtain the desired probability. 12 CHAPTER 1. DISCRETE PROBABILITY ... Pro-gramming, vol. II (Reading: Addison-Wesley, 1969). 30 CHAPTER 1. DISCRETE PROBABILITY DISTRIBUTIONSHistorical RemarksAn interesting question in the history of science is: Why was probability...
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Introduction to Probability - Chapter 2 docx
... 355 /113 :L =πP(E)2=12355 113 18083408=56= .8333 .Even with careful planning one would have to be extremely lucky to be able to stopso cleverly.The second author likes to trace ... referring to a continuous random variable X (say with a uniform densityfunction), it is customary to say that “X is uniformly distributed on the interval[a, b].” It is also customary to refer to the ... Gridgeman comments that Lazzerini’s estimate turnedout to agree with a well-known approximation to π, 355 /113 =3.1415929, discov-ered by the fifth-century Chinese mathematician, Tsu Ch’ungchih. Gridgeman...
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Introduction to Probability - Chapter 3 ppt
... =12342143,indicating that a1went to a2, a2 to a1, a3 to a4, and a4 to a3.Ifwealwayschoosethetoprowtobe1234then, to prescribe the permutation,we need only give the bottom row, with the understanding ... Birthday problem.We now turn to the topic of permutations.PermutationsDefinition 3.1 Let A be any finite set. A permutation of A is a one -to- one mappingof A onto itself. ✷ To specify a particular ... sent by the one -to- one mapping. For example, if A ={a, b, c} a possible permutation σ would beσ =abcbca.By the permutation σ, a is sent to b, b is sent to c, and c is sent to a. Thecondition...
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Introduction to Probability - Chapter 4 doc
... drawn is red? 4.3. PARADOXES 17513 Write a program to allow you to compare the strategies play-the-winner andplay-the-best-machine for the two-armed bandit problem of Example 4.24.Have your program ... Example 4.2, we used the Life Table (see Appendix C) to compute a con-ditional probability. The number 93,753 in the table, corresponding to 4 0- year-old males, means that of all the males born in ... find the probability that amale of age 60 in 1981 lives to age 80. Find the same probability for a female.32 (a) There has been a blizzard and Helen is trying to drive from Woodstock to Tunbridge,...
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Introduction to Probability - Chapter 5 docx
... appears to grow without limit.We can now ask: How long will a customer have to wait in the queue for service? To examine this question, we let Wibe the length of time that the ith customer has to ... How long shouldeach customer expect to wait? (We define the waiting time of a customer to be thelength of time between the time that he arrives and the time that he begins to beserved.)Let us ... distribution to X? If it is appropriate,what probability should we assign to each outcome?(a) Take the first 100 students who enter the cafeteria to eat lunch.(b) Ask the Registrar to sort the...
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Introduction to Probability - Chapter 6 doc
... program to compare the variances for thefollowing densities, both having expected value 0:pX=−2 −10123 /11 2 /11 1 /11 2 /11 3 /11 ;pY=−2 −10121 /11 2 /11 5 /11 2 /11 1 /11 . 256 CHAPTER ... from the definition of V (X). 236 CHAPTER 6. EXPECTED VALUE AND VARIANCEWLWLWLWLWLWL(2,3,12) L1098654(7 ,11) W1/32/32/53/55 /11 6 /11 5 /11 6 /11 2/53/51/32/32/91/121/95/365/361/91/121/91/362/362/453/4525/39630/39625/39630/3962/453/451/362/36Figure ... pairs has the same probability. Then the player wins in the first five cases and loses in the last six.Thus the probability of winning is 5 /11 and the probability of losing is 6 /11. Fromthe path...
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Introduction to Probability - Chapter 7 pptx
... variance 1, then (cf. Exam-ple 7.5)fXi(x)=1√2πe−x2/2,4J. B. Uspensky, Introduction to Mathematical Probability (New York: McGraw-Hill, 1937),p. 277. 290 CHAPTER 7. SUMS OF RANDOM ... series lengths: four-game, five-game, six-game, seven-game.(b) Using one of the distribution found in part (a), find the probability thathis batting average exceeds .400 in a four-game series. (The ... customer at a bank is exponentially dis-tributed with mean service time 2 minutes. Let X be the total service timefor 10 customers. Estimate the probability that X>22 minutes. 288 CHAPTER...
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Introduction to Probability - Chapter 8 docx
... <n,orSnn<.We wish to show that the probability of this event tends to 1 as n →∞,or equivalently, that the probability of the complementary event tends to0 asn →∞. The complementary ... x) to compute the exact probability that you estimated in Exercise 1. Compare the two results.3 Write a program to toss a coin 10,000 times. Let Snbe the number of headsin the first n tosses. ... 65 and 75. 11 The Pilsdorff beer company runs a fleet of trucks along the 100 mile road fromHangtown to Dry Gulch, and maintains a garage halfway in between. Eachof the trucks is apt to break down...
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Introduction to Probability - Chapter 9 pps
... answer in this case, to five places, is .99821. ✷It is important to work a few problems by hand to understand the conversionfrom a given inequality to an inequality relating to the standardized ... first time the total sum of the face values of the dieis 700 or greater. Estimate the probability that, for this to happen,(a) more than 210 tosses are required.(b) less than 190 tosses are required.(c) ... Theorem, to estimate the prob-ability that S lies between 499,500 and 500,500. Use the same two methods to estimate the probability that S lies between 499,000 and 501,000, and theprobability...
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