sMT1SOL_06S - Question 1(15 A six—faced die with faces...

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Unformatted text preview: Question 1: (15%) A six—faced die, with faces numbered as 1, - - - ,6, is tossed and the number of dots facing up is noted. ‘ 1. (5%) Show that the event A = {X Z 4} “implies” B = {(X +4) is not a prime number} “implies” C' = {X 2 2}. 2. (5%) Construct the probability of the elementary events under the assumption that the face with a single dot is three times as likely to be facing up as any one of the rest 5 faces. 3. (5%) Under the above probability assignment, find the probabilities of P (A), P(B) P(C). 1- 24:?45‘6‘5) B=f2)4)5,e)) c=f2,3,4,5,65 Therefwe AgBQC. 7 Pacg=a11345é :3 306+?C+?C+‘7C+7C<~7C We‘gkt 395 7c 96 7C 76% - _ :_L 4&4, 76 g Question 2: (10%) 1. (3%) Using the Venn diagram to prove the DeMorgan’s Rule: For all events A, B Q 3, (A U B)c 2 AC fl BC. 2. (3%) Using the first result, prove (AUBUCUD)C=AanchCnDC. 3. (4%) Throw a fair die four times, and assume each die—tossing is independent. Corn— pute the probability that a single dot is facing up at least once. ACOBC, 2‘ “First FHA/9 . (MUG: @usfnccz Am 13ch (AUBUCUD)Q=(AUBUC)C(1 DC == ACflBCfl Ccfl DC 3‘ P< a {pale olO‘t ‘Paees up 0:6 least Ghee) :5. {>01 Srnék 015m NEVER fines up) = l-- (S 4“ 54’54 __ (97’ ~.. 54 me “then Question 3: (7%) A random experiment has a sample space S = {m, y, 2}. Suppose that P({$, 2}): 1/3 and P({y z})— — 7/9. Use the axioms of probability to find the probabilities of the elementary events, namely, What are P( (,{x}) P({y}), P.({z}) We have P(f%})+ o +p<fgi>~elg=flfmw P(l‘éilHPClEl): :5}: Ftihfifil) Question 4: (12%) Two numbers X and Y are independently selected from the interval [0,1] uniformly randomly. 1. (5%) Find the probability that they differ by more than 1/3. 2. (7%) The two events A and B are defined as follows: A B {X > 0.5}, {X > Y}. Show that A and B are NOT independent. % . PCAflB)2‘§-*P6Al PCB) g A as are, NOT New; Question 5: (17%) A normal six—faced fair die is tossed and the number of dots N1 is noted; an integer N2 is then selected from {N1, - - - ,6} uniformly randomly. 1. (2%) Specify the sample space. 2. (5%) Use either a tree diagram or a table method to construct the weight assign— ments (the probabilities). 3.5( %) Find the probability of the event {N1 + NZ > 10} (5%) Find the probability of the event {N12 4} given {N2— — 6} /. N> 2 3 <1. 56 $111,041», (1.3),114H15) I 11, ' x .. I ' (>1>>1C>3),(2\4) [25) C2£ WM 5 I 2 i hiifivagfifi (3’3” 861)» 8568-6) 4 11111 111‘ 11 Isa “M44950 5 11111111111111111 1 “516-16) 1; llllllllil l (616); 1 A: finite Ohm/e “(6pr J g /\_,66 :5" { Sum up “Hie ‘FOW’ block 4 Paw-11A >10) :51. +19% 38-ka .1. .1 3 ’C G reel—aesxe 66+ 161: _. 32 1, 2 l4 " M7 Question 6: (15%) A salesperson travelled between cities A, B, and C' for four ’ consecutive nights, and each night he could only stay in one city. At day 1, the salesperson started from city A and stayed there for the first night. For the next three mornings, the salesperson uniformly randomly selected the next destination from the two cities excluding the city he stayed for the last night. For example, if he stayed in city B for the third night, the fourth night of his can only be in either city A or city 0. 1. (8%) What is the probability that the salesperson was able to Visit all three cities? 2. (7%) What is the probability that the salesperson visited city A twice, given that the salesperson had visited all three cities. 3L. d; l J- i 3 i 12— E C A B c A c 25’ ‘3 “:23 :‘Z "" 3 3 , \ PC VEEC all CEUQS‘ ): -6K(é) ‘5- 2: all Frobdhxlrtej A under which \ ll there (S 0x “2:: H A F: vetted mice “a“- % all cities we, visited) :Arr-l-i-Tzi: ou prob. MUN 813% i 3: 3 w 4. 3&2 \3 PM is View mice \ all cares are meat): Question 7: (24%) The waiting time X of a customer in a queueing system is zero if he finds the system idle, and an exponentially distributed random length of time if he finds the system busy. The probabilities that he finds the system busy or idle are q and 1 — g, respectively. 1. (2%) What is the sample space? Hint, the waiting time t takes values in real numbers. 2. (7%) What is the corresponding cumulative distribution function (cdf)? Hint: con— sider two cases: at < O and :r 2 0. 3. (3%) Using the cdf obtained in 2., compute the probability P(X E [0, 3])? 4. (2%) What kind of random variables is X? A discrete random variable, a continuous random variable, or a mixed random variable? It is a multiple—choice question. No need to write down the justification. 5. (10%) Construct the generalized probability density function (pdf) using delta func— tions, namely, using 6(51: — 2:0). For your reference, the pdf of an exponential distribution is f (t) = Ae‘“, where A > 0 is a parameter. " i idle, husyfi >5 H? or gym can write fwflg) @333 X [0) DO) FxCxlzq O ac , M13721: + (hprl = 3w 8.1%)1'Chg) -.\___fl__r_,..,_..7 W first Wm Carrestk 3“?) [X6 [031)2363) ~FK CO;)‘ .170 The case film W ._ l"%€ Misfire Case—Chit 4. Mid. Shave, We is Q gump Wm Eco—>30 %E<(0)=l»fy 5. My “the gimp Six do fifiPeIem‘tgdtfm. \ flock Sontag.) + { 233?);ng long?) he x30 Question 8: (15%) Suppose X is a random variable uniformly distributed on [—1,1], and Y is defined by Y 2 X72 +1. 1. (8%) Find the cumulative distribution function Fy(y) = P(Y S y). Hint: consider two cases: y < 1 and y 2 1. 2. (7%) Find the probability density function fy (y). JFK-Jim-J '— '-~:T:.——’.'w—- —-v.'t:='1 :. A _— . ._ __ ‘ ' ._ I‘. . JII‘QA‘iL/A‘vfl! l?£.'.£!;l.5?“"‘€u""'" ‘ W W (62$ ' 2 PCB?» ...
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