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Berkeley STAT 134 - Homework 7

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Statistics 134: Concepts of Probability (Lugo)Homework 7This assignment is due on Friday, October 21, in class. In order to do it you should readSections 3.5 and 3.6.Recommended problems‘ Section 3.5: 1, 3, 5, 7, 9, 11, 13, 15, 17.Section 3.6: 1, 3, 5, 7.Chapter 3 review: 25.If you’re interested in the theory, 3.5.19 and 3.6.11, 13, 15 are also nice problems – butthey’re a bit too hard for this course.(Remember: you don’t need to hand these in. Also, I realize that the “recommendedproblems” lists have gotten a bit long lately. You should really be interpreting these as“problems you should be able to do” – it’s not necessary that you actually do all of them.But I do recommend at least reading the statements of them and thinking for a momentabout how you would do them.)Required problemsPitman 3.6.8 A deck of 52 cards is shuffled and split into two halves. Let X be the numberof red cards in the first half. Find:(a) a formula for P (X = k)(b) E(X)(c) SD(X)(d) P (X ≥ 15) approximately, using the normal curve.Pitman 3.6.9 A population contains G good and B bad elements, G+B = N. Elementsare drawn one by one at random without replacement. Suppose the first good elementappears on draw number X. Find simple formulae, not involving any summation from 1 toN, for:(a) E(X)(b) SD(X).[Hint: Write X − 1 as a sum of B indicators.]This is an odd-numbered problem. Feel free to look in the back of the book for the answer.The point of the problem is really to derive the answer.P27: misprints Books from a certain publisher contain an average of 2 misprints perpage. What is the probability that on at least one page in a 250-page book from this publisherthere will be at least 6 misprints?1P28: independent Poissons Suppose X, Y, Z are independent Poisson random vari-ables, each with mean 2. Find:(a) P (X + Y = 5) (b) E((X + Y )2) (c) P (X + Y + Z = 7).P29. A deck of 52 cards is shuffled and dealt. Find the probabilities of the followingevents:(a) the twelfth card is a jack;(b) the last four cards are hearts;(c) the twelfth card is a spade;(d) the last jack appears on the 49th card.Now call the events of (a), (b), (c), (d) A, B, C, D respectively. Find:(e) P (C|A)(f) P (C|B).P30: records in a sequence Suppose 52 cards numbered 1 to 52 are shuffled and dealtone by one.(a) You receive one cent for the first card and then one cent for each card dealt whosenumber is smaller than those of all previous cards dealt. What are your expected winnings?Consider, for example, the six-card case. If you’re dealt 3, 5, 2, 6, 1, 4 in order, then you win onecent for each of 3, 2, and 1, for a total of three cents. Hint: what’s the probability that you winone cent when the card i is dealt, for each i?(b) Assume that the variance (measured in squared cents) of your winnings is the sameas the expectation, measured in cents. (This is approximately true; I’ll try to give a proof ofthis in the solutions.) If you paid six cents for each play of this game, and played 25 times(meaning you paid a total of 150 cents for 25 separate deals of the deck) what, approximately,is the chance that you would come out ahead?P31. A mixture of two Poissons. Let X be a Poisson(λ) random variable, and letY be a Poisson(µ) random variable. Let Z be a random variable constructed as follows: flipa fair coin. If it comes up heads, then let Z = X; if it comes up tails, then let Z = Y .Observe that the generating function of a Poisson random variable with mean λ isXke−λλkzkk!= e−λXk(λz)kk!= e−λeλz= eλ(z−1).(a) What is P (Z = k)? (You don’t need the generating function for this.)(b) What is E(Z)?(c) What is V ar(Z)?(d) For which choices of λ and µ is V ar(Z) = E(Z)?In those cases in (d), and in no others, Z is itself Poisson.(Hint: for (b) and (c) the best way to proceed is to derive a formula for the mean andvariance of Z which is constructed from any random variables X and Y in this way, andthen specialize to the


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Berkeley STAT 134 - Homework 7

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