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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are withdrawn from the urn.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=iX+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(3l)=§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[XY = 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 EXY =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=_\~= liXl I‘orally 10. Suppose X and Y are independent continuous random variables. Sht)‘ liXY=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[X2Y = 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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 Spring '08
 Lim
 Operations Research

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