Scan HW1 - Exercises 91 40. Suppose that two teams are...

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Unformatted text preview: Exercises 91 40. Suppose that two teams are playing a series of games, each of which is independently won by team A with probability p and by team B with probability 1 — p. The winner of the series is the first team to win four games. Find the expected number of games that are played, and evaluate this quantity when p = 1/2. 41. Consider the case of arbitrary p in Exercise 29. Compute the expected num- ber of changeovers. 42. Suppose that each coupon obtained is, independent of what has been previ- ously obtained, equally likely to be any of m different types. Find the expected number of coupons one needs to obtain in order to have at least one of each type. Hint: Let X be the number needed. It is useful to represent X by m X=ZX,~ i=l where each X i is a geometric random variable. ‘ 43. An urn contains n + m balls, of which n are red and m are black. They are withdrawn from the urn. one at a time and without replacement. Let X be the number of red balls removed before the first black ball is chosen. We are interested in determining E [X ]. To obtain this quantity, number the red balls, from 1 to n. Now define the random variables X ,-, i = l, . . . , n, by X_ _ 1, if red ball 1' is taken before any black ball is chosen ‘ - 0, otherwise (a) Express X in terms of the X ,-. (b) Find E [X]. 44. In Exercise 43, let Y denote the number of red balls chosen after the first but before the second black ball has been chosen. (a) Express Y as the sum of n random variables, each of which is equal to either 0 or 1. (b) Find E[Y]. (c) Compare E [Y] to E [X] obtained in Exercise 43. (d) Can you explain the result obtained in part (c)? 42/A total of r keys are to be put, one at a time, in k boxes, with each key ' dependently being put in box i with probability pi, {5:1 p,- = 1. Each time a key is put in a nonempty box, we say that a collision occurs. Find the expected number of collisions. Exercises 165 . [— lJlla. Pr+.(l> + firm-“(0’1 P (2) = r 2p . l _ 7 - l = 171%th I)p’+'(l — WNW/"+11 -p r(l — l P,-(3) = ’zp—p—[al P,.+1(2) + 2012 Pr+l(1) + 3013‘?“ ‘0’] and soon. Exercises 1. If X and Y are both discrete, show that Z“. pxlyuly) = l for all y such that PYO‘) > 0. *2. Let X 1 and X 3 be independent geometric random variables having the same parameter [1. Guess the value of PlX1=i|X|+X2=n} Hint: Suppose a coin having probability p of coming up heads is continually flipped. 1f the second head occurs on flip number n, what is the conditional probability that the first head was on flip number i, i = l. . . . , n —’l? Verify your guess analytically. 3. The joint probability mass function of X and Y. p(x. y). is given by l _. P(3-l)=§y Compute ElXIY =i] fori = l.2.3. 4 In Exercise 3. are the random variables X and Y independent? /5/ An urn contains three white, six red, and five black balls. Six of these balls are randomly selected from the urn. Let X and Y denote respectively the number of white and black balls selected. Compute the conditional probability mass function of X given that Y = 3. Also compute E[X|Y = l]. *6. Repeat Exercise 5 but under the assumption that when a ball is selected its color is noted, and it is then replaced in the urn before the next selection is made. 1 66 3 Conditional Probability and Conditional Expectation 7. Suppose [)(.\'. .r. 1). the joint probability mass I'unction of the random ables X. Y. and Z. is given by no.1 l)=;L-. p(2.I.l)='I ptl l 2):? [7(212)=%. What is E|X|Y =2]? What is EleY =2. 2 = II? 8. An unbiased die is successively rolled. Let X and Y denote. l‘t tively. the number of rolls necessary to obtain a six and a five. Find (a) (b) EIXIY = II, (c) EleY =5]. 9. Show in the discrete case that if X and Y are independent. then liIXlY=_\~|= li|Xl I‘orally 10. Suppose X and Y are independent continuous random variables. Sht)‘ li|X|Y=y|=EIXl l‘orallj‘ 11. The joint density of X and Y is 1 ‘3 (V —.\“) ' t' ‘. ()<.\‘<oo. —.\'<.\'<.\' 8 Show that EleY = .\-| = 0. 12. The joint density of X and Y is given by ftx. y) = e—‘x/ye"). 0<.\' <00. 0<.\~ <oo f(x. y) = .V Show EIXIY = _\‘l = y. *13. Let X be exponential with mean UK: that is. .l'x(.\') = AF)”. () < .\' < 00 Find EleX >11. 14. Let X be uniform over((). I). Find EleX 4 ill. 15. The joint density of X and Y is given by )_.\. _f'(.r.)‘)=‘—f. 0<.\' <y. ()<_\'<oo ‘\ Compute E[X2|Y = y]. W_.. = 168 3 Conditional Probability and Conditional Expectation (a) The X; are normal with mean 0 and variance 1. (b) The density of X,- is f(x) = Be’e", x > O. (c) The mass function ()in is p(x)=l9"(1— 9)""', x =O,1, 0 < 6 <1. (d) The X ,- are Poisson random variables with mean 0. *19. Prove that if X and Y are jointly continuous, then 00 E[X] = f E[XIY = .vlfy(,v)d.v —OO 20. An individual whose level of exposure to a certain pathogen is .r will con— tract the disease caused by this pathogen with probability P(x). If the exposure level of a randomly chosen member of the population has probability density func- tion f, determine the conditional probability density of the exposure level of that member given that he or she (a) has the disease. (b) does not have the disease. (c) Show that when P(x) increases in x, then the ratio of the density of part (a) to that of part (b) also increases in x. 2 onsider Example 3.12 which refers to a miner trapped in a mine. Let N enote the total number of doors selected before the miner reaches safety. Also, let T,- denote the travel time corresponding to the i th choice, i 2 1. Again let X denote the time when the miner reaches safety. (a) Give an identity that relates X to N and the T,~. (b) What is E[N]? (c) What is E[TN]? (d) What is 12123,”:l T,-|N = n]? (e) Using the preceding, what is E [X ]? 22. Suppose that independent trials, each of which is equally likely to have any of m possible outcomes, are performed until the same outcome occurs k consec— utive times. If N denotes the number of trials, show that mk—l E [N] = m — 1 Some people believe that the successive digits in the expansion of I! = 3.14159 . . . are “uniformly” distributed. That is, they believe that these digits have all the appearance of being independent choices from a distribution that is equally likely to be any of the digits from 0 through 9. Possible evidence against this hypothesis is the fact that starting with the 24,658,60lst digit there is a run of nine successive 7s. Is this information consistent with the hypothesis of a uniform distribution? To answer this, we note from the preceding that if the uniform hypothesis were correct, then the expected number of digits until a run of nine of the same .._—_:-=-u=n=b—_=.=L_' Exercises 169 value occurs is (l09—l)/9=lll.lll.l1l Thus, the actual value of approximately 25 million is roughly 22 percent of the theoretical mean. However. it can be shown that under the uniformity assump- tion the standard deviation of N will be approximately equal to the mean. As a result, the observed value is approximately 0.78 standard deviations less than its theoretical mean and is thus quite consistent with the uniformity assumption. *23. A coin having probability p of coming up heads is successively flipped until two of the most recent three flips are heads. Let N denote the number of flips. (Note that ifthe first two flips are heads, then N = 2.) Find E[N]. 2 A coin, having probability p of landing heads. is continually flipped until at east one head and one tail have been flipped. (a) Find the expected number of flips needed. (b) Find the expected number of flips that land on heads. (c) Find the expected number of flips that land on tails. ‘ (d) Repeat part (a) in the case where flipping is continued until a total of at least two heads and one tail have been flipped. gm gambler wins each game with probability p. In each of the following cases. determine the expected total number of wins. (a) The gambler will play n games; if he wins X of these games,.then he will play an additional X games before stopping. (b) The gambler will play until he wins; if it takes him Y games to get this win, then he will play an additional Y games. 26. You have two opponents with whom you alternate play. Whenever you play A, you win with probability pA; whenever you play B. you win with probability pg, where p B > [)A. If your objective is to minimize the number of games you need to play to win two in a row, should you start with A or with B? Hint: Let E [N,-] denote the mean number of games needed if you initially play i. Derive an expression for E [NA] that involves E [NB]; write down the equivalent expression for E [N B] and then subtract. 27. A coin that comes up heads with probability p is continually flipped until the pattern T, T. H appears. (That is, you stop flipping when the most recent flip lands heads, and the two immediately preceding it lands tails.) Let X denote the num- ber of flips made. and find E[X]. 28. Polya’s urn model supposes that an urn initially contains r red and b blue balls. At each stage a ball is randomly selected from the um and is then returned along with m other balls of the same color. Let X k be the number of red balls drawn in the first k selections. ...
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Scan HW1 - Exercises 91 40. Suppose that two teams are...

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