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c any? ref/W: ﬂ @égﬁwg' We .. STAT 333 Spring 2006 Term Test #2 July 5, 2006, 4:305:25 pm. Nunez”—
Instructor: C.D. Cutler LBJL ? Instructions 0 Please be sure to enter your NAME and student ID. at thetop of this page.
0 The test has 5 pages. Please check that you have all the pages.
0 Calculators may be used. No other aids are permitted.  0 Part A of the test consists of 7 multiple choice questions of equal value. For each question
select the one BEST answer. When you complete Part A, please enter the letters corresponding to your answers in the
blanks provided at the end of the section. Incorrect answers score 0. 0 Part B consists of 2 long—answer problems. Please answer these in the space provided. Marking Scheme: Part A: 20 Part B:
1. 10
2. 12 + 3 = 15 Total: 45 Part A Multiple Choice t 1. A coin is biased with probability p = 1 / 4 of turning up Heads. The coin is tossed repeatedly until 3 Heads are obtained. Let N be the number of tosses to obtain the 3 Heads. Now
suppose the coin is tossed N times and X = number of Heads in those N tosses. Then E(X)
is: (a) 3 (b) 12 (c) 4 (d) 1‘s .(e) 6 g g 2. Digits are picked randomly with replacement from the set {0, 1, 2,... , 9}. Find the expected
waiting time until the ﬁrst occurrence of the sequence “9 0 9 9 1 9 O 9 9”. (a) 105 I 7' J m (b) 104 + 109 ”7" 7’ 6? )10 + 104 + 109 V / J /l
(d) 102 + 104 + 109 /f‘”ﬁ’ 95’” 4 77
(e) 109 9 /i) *‘f' (/ﬂyl’f /p 3. Let A be a renewal event. Let T), denote the waiting time until the ﬁrst occurrence of A, let
f; = P(T)‘ < co), and let rn, n = 1,2, . . . denote the renewal sequence. Suppose N A is the
total number of occurrences of A. Then A is transient if and only if ,KIKNA) < 00 (b) 2:11” < oo
(o) f), < 1 and 2:115:00 @wmm (e) none of the above 4. Suppose A is a delayed renewal event with associated renewal event A. Suppose the ﬁrst
waiting time distributions of A and A have respective generating functions: 3
1N$=rn%3§ “d ﬁw=ITdiﬁ Then the expected number of steps required to see 4 occurrences‘of A is: (3)4 (b) 12 @10 (d) 15 (e) 20 / 7/; +51 / .C‘ ,(«Z+2J) 33 $7 0
[6/ f: — W ......... . 3 ..”_.+ .. :3. I I; 3 Hm) 3/0 5. Jane and John are playing a series of card games. The games are independent, and on each M game Jane’s probability of winning is p = 0.6. Suppose Jane starts with $50, John starts with $100, and on each game the loser pays the winner $10. Find the probability that Jane
goes bankrupt before John: (a) 0.3182 (b) 0.2163 (c) 0.9850 (d) 0.8703 .1297
$414 ti MM % /,.. (3/2.) {0 l
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1 6. Consider an ordinary random walk on the integers where the probability of a jump to the
right is p: 0.3. The probability of ever reaching state 4 IS: 1/
(a) .008 . .034 (c) .134 (d) .429 (e) 1 Z (afar?) ® 7. Refer to the above random walk problem. Starting from the origin, the expected number of
visits to state 2' is: (a) 10.5 (b) 6.5 (c) 4.5 (d) 2.5 . (e) 1 _Z/ r f 7 f: //” 1 4 Q1), 60
awn.) 1 0,4
0 if [ﬁjﬂukw M ): T / a;
(£3) Please :6K§:T LETTERS :espE/mrojéj uranswers in the blanks below: lg. Part B Long Answer 1. Suppose we have a sequence of coin tosses Where the probability of Heads is 0 < p < 1. Derive
the probability generating function F(s) of the waiting time T until the ﬁrst occurrence of
the event ”H T H”. (Show your work). Then obtain E(T) using any valid method. 3% I‘J/z‘gw fly/”é 3657?“; f I {Wffyf’
F/ﬁ #(gﬁﬁvfﬁf 5 ”W Z at 2. Suppose we have a modiﬁed random walk with a reﬂecting barrier at state 0 where the state
space is {. . ., —2, —1,0} and the walk acts like an ordinary random walk (jumping to the right
with probability p > 0 and to the left with probability q = 1 — p > 0) when it is in any state
other than k = 0. However, if the walk hits 0, at the next step it jumps backward to state —1. (a) Determine the values of p (0 < p < 1) for which “return to 0” is transient, null recurrent,
and positive recurrent. Provide an argument for your answer. (You may use known results
from the ordinary random walk.) fill: V? WWW“ . ‘ﬁ 9‘2; ’4”an [f "531:“le
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é; ' ' ‘ A, ,. Q (Mk) 1 Ct
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Q . ”AWWWwwmm , , V l (b) Let Too = number of steps between returns to 0. Suppose p = 0.6. Find E(Too). 21¢“, . /
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 Chisholm

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