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Berkeley STAT 134 - Homework

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Statistics 134: Concepts of Probability (Lugo)Homework 4This assignment is due on Friday, September 23, in class. In order to do it you should readSections 2.1, 2.2, and 2.4. (I’m not going to talk about Section 2.3 in class. This sectionderives the normal approximation to the binomial distribution from consecutive odds ratios.If this sounds interesting to you, read it; I’ll be glad to answer any questions you mighthave.)The first exam is on Wednesday, September 28, and it covers Chapters 1 and 2. Homework5 will be due on Friday, October 7; it will be a bit longer than usual. It will include somerecommended problems from Section 2.5 and/or from the Chapter 2 review problems thatyou should do before the exam; I’ll make sure to point out which ones those are.Recommended problemssection 2.1, problems 3, 7, 11.section 2.2, problems 3, 9.section 2.3, problem 2. (This is in 2.3, yes, but it doesn’t need to be there.)section 2.4, problems 3, 7, 9.Chapter 2 Review, problems 1, 5, 7, 9, 11, 19, 25, 27.(Remember: you don’t need to hand these in.)Required problemsPitman 2.2.8 Find, approximately, the chance of getting 100 sixes in 600 rolls of a die.(We’ve talked about multiple ways to approximate this; any of them is fine.)Pitman 2.2.14 Wonderful Widgets Inc. has developed electronic devices which workproperly with probability 0.95, independently of each other. The new devices are shippedout in boxes containing 400 each.(a) What percentage of boxes contains 390 or more working devices?(b) The company wants to guarantee, say, that k or more devices per box work. Whatis the largest k such that at least 95% of the boxes meet the warranty?Pitman 2.4.10 Let N be a fixed large integer. Consider n independent trials, each ofwhich is a success with probability 1/N. Recall that the gambler’s rule (see Example 1.6.3)says that if n ≈23N, the chance of at least one success in n trials is about 1/2. Show that ifn ≈53N, then the chance of at least two successes is about 1/2.Remark. The constant 2/3 is a good approximation of log 2 ≈ 0.693. In general, we canfind some constant n = αkN such that the chance of at least k successes in n trials is about1/2. So α1= log 2 ≈ 2/3 (the gambler’s rule) and this problem asks you to show α2≈ 5/3.It turns out that in general limk→∞(αk− k) = −1/3 – that is, αk≈ k − 1/3 – but I don’tknow a simple proof of this.MORE PROBLEMS ON NEXT PAGE1P12: Conditional probabilities in shootingA man fires 9 shots at a target. Assume that the shots are independent, and each shothits he bull’s eye with probability 0.6.(a) What is the chance that he hits the bull’s eye exactly 4 times?(b) Given that he hits the bull’s eye at least twice, what is the chance that he hit thebull’s eye exactly 4 times?(c) Given that the first two shots hit the bull’s eye, what is the chance the he hits thebull’s eye exactly 4 times in the 8 shots?P13: Time until the the nth successI roll a die until the numbers 1 or 2 have appeared a total of six times. For example, if Iroll 5, 1, 6, 2, 4, 5, 1, 3, 6, 2, 4, 3, 5, 1, 2 this requires fifteen rolls.(a) What is the probability that I have to make exactly fifteen rolls?(b) What is the probability that I have to make at least sixteen rolls?P14: Normal approximationsLet H be the number of heads in 100 tossess of a coin which is biased with probability0.51 of coming up heads. Find:(a) P (45 ≤ H ≤ 55). (b) P (55 ≤ H ≤ 60). (c) P (H = 50). (d) P (H = 60).P15: Draws from a large boxA box contains 1000 balls, of which 3 are black and the rest are white.(a) Which of the following is most likely to occur in 1000 draws with replacement fromthe box: fewer than 3 black balls, exactly 3 black balls, or more than 3 black balls? (Hint:this sounds like a conceptual problem but it’s really not. Compute the probabilities.)(b) If two series of 1000 draws are made from this box, what is the approximate probabilitythat they yield the same number of black

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