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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£
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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) :ArrliTzi: 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
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= 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). JFKJimJ '— '~:T:.——’.'w— —v.'t:='1 :. A _— . ._ __ ‘ ' ._ I‘. .
JII‘QA‘iL/A‘vﬂ! l?£.'.£!;l.5?“"‘€u""'" ‘ W W (62$
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 Spring '08
 GELFAND

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