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Unformatted text preview: Theory of Probability
Prof.Dr.M. Nahit Serarslan
Res. Assist, H. Kutay Ting, Res. Assist. Evsen Korkmaz Res. Assist. Pinar Dursun. Res. Assist. Ronay Ak
25.05.2009 Final Exam
Duration: 150 rnin
 Please show all your work and use verbal explanations, if necessary. 1. Suppose that before observing new evidence the hypothesis H is three times more likely to be
true as is the Hypothesis 6. lfthe new evidence is twice as likely when G is true thanit is when H is 10 pts true, which hypothesis is more likely after the evidence has been observed? PM): 3/98) (net Swine 9;: a.) Praia): 2. P(ElH) (gown where.) P[
P/H/E) : ﬂ _ PKHE) _ PUMP/EM) P/QIE) Wee) PréE) _ r76). P/Elé‘) Hence , Al ,3 more L249 aﬂer‘ +Ae €Vr’oﬁzOCz has been
05W. Final — Page 1 of a certain type is Poisson random
be observed if it is
the leaf. if we let Y an insect gs laid on a tree leaf by
h a random variable pan only 2, The number of eg
r, suc iable with parameter A. Howeve var
0 then we cann positive, since if it is
e the observed number of eggs, P{Y=i)=P{X=ilX>0} denot then w . . .
here X IS Paisson wuth parameter/l. Find E[V] Pfyz [I X>o§ = ?iX:L',X>0i P{X>o§ EM= :— mas}
4” P§Xyoj = Efxl
ﬁve? A —' _———.— 1— gr? Final # Page 2 3. The following dartboard is a square whose sides are length 6. The three circles are all
centered at the center of the board and are of radii 1, 2 and 3. Darts landing within the circle
of radius 1 score 30 points, those landing outside this circle but within the circle of radius 2
are worth 20 points, and those landing outside of circle radius 2 but within the circle of radius
3 are worth 10 points. Darts that do not land within the circle of radius 3 do not score any
points Assuming that each dart that you throw will, independently of what occured in your previous throws, land on a point uniformly distributed in the square, find the probabilities of
the following events. a) You score 20 on a throw of the dart.
b) You score at least 20 on a throw of the dart. 12 pts
c) You score 0 on a throw of the dart. d) The expected value on your score on a throw of a dart
e) Both of your first two throws score at least 10. f) Your total score after two throws is 30. N Area (s.)=36»9r (0 M) Lei Xc'denoié your scan: on ,‘Jh +6rou) [4,: 1,1,...) 0 X: :31’ 1: ' ;3. . 40.21
)PE .20} 36— ,4 0/) thJ 03%;,20335», 36 34 ’ '3— 9 . y
a) P5X.=o} = 35;.” , ,_ 1: a) Piwg2rols/I—F{)¢.o}),(4ﬁgxfo}
a q 2. 
O _ YrsFl) 4—291
F) ﬂaw/”530}  2.(P§X=30,x4.+1:o}+ M120“ “4) 5/95/9320}; 3r+F  7r = 357: If] =2. l/7‘§—)+L.§I 36 49. 26 5. If 2 is a standard normal random variable, what is COV (Z, 22)?
' ' 10 pts Cov my) : E. [(XEch)(v 500)] COV (93,12): E[(2EC23) (%‘—EC%‘3)] Near) 049/ Mob/,0, 0,5 mag/ad normal
random VCm'ob/e at 0 000/ f
M { ,
Way. Vern?) = ECZ‘J {£1301
4 = E5217 ~ 0 Eff] :1 =>Cov (2,?2)= E[% (22.4)] _ N 0
: tCa‘j
l. 7— [5&3]: 23._L__¢'2%o/2— ,90 V517 1489’ij mh‘jm‘hon by [3015, E‘Jj usiry)
_ 1.7:
ﬂea 2 e zen/2'0” . Finalm Page 5 6. The joint density ofXand Y is given by 2x dSy
raw—‘65 Cl<x<eoo<y<oo 10 t5
0 of 'zeruﬂse p Find the density function of the random variable X/Y. F(>(/\/)= P{X/‘/ gag Off 5 6.29% 33mg =]( 361 ~2’Xe SJCB d3 Final — Page 6 7. The neighborhood bank has three tellers, two of them fast, one slow The time to assist a
customer is exponentially distributed with parameter r). = 5 at the fast tellers, and = l~ at the slow teller. Jane enters the bank and chooses a teller at random, each one with 10 pts
probability 1/3. Find the PDF of the time it takes to assist Jane and the moment generating
function 31:?) L51" X (21.20046 Sen/Ace #er / \/ demo—[c Adler; ﬁx m = 2‘. _€,y{"/y) WM} VJ Harm/74' jeﬂmb’y Anchbj of XI 00 44%
4 ‘ 04X
Mbk): Efefyffaty/é. 66 ’73:?ch Final — Page 7 8, Your company must make a sealed bid for a construction project. if you succeed in winning
the contract (by having the lowest bid), then you plan to pay another firm 100 thousand
dollars to do the work. lf you believe that the maximum bid (in thousands of dollars) of the
other participating companies can be modeled as being a value of a random variable that is 10 pts uniformly distributed on (70, 140), how much should you bid to maximize your expected
profit? [F JO” 5’5! 70 yoéaré/l/O , Ham you W'” W,” +hc 5'6 0”!” Make a foroFH— 070 ar— #00 W’V’Lh
imbue/"w {ma—«Mm ) Or my, +A¢ 5:2 we Page
5""?! (”who and more mm 0 MM «>70 0.
HenC€ ,UOUF CXPCCJCDI [04974; v13, —4—— (a {00).(4140 x) 3 3L0 (2mm  93— 414000) _4._
4O
D’iL/‘é’mh’af—lty 000/ Sci—H29 1‘41"— [OfeCéeol'l’B
479 O ja'VES lhO—lx; O Winona / you should 590/ 0.0 'Ilhoujono/ c/o/lo/S: Your expgcko/ PM 72+ W” #120 A; 4/0/7— muse/m]
elollarS. Final — Page 8 9. The joint density of X and Y is given by folI) = {xe‘o‘fﬂ x > 0,); >0 0 o {Faer351: a Are X and Yindependent? 4mg)
~ )
£/%)=fm”w = 0
O
— 7( é’Xe'Oo— [—X 65“)
: Ke'k
9° —{9r+) 4””
79(7): fate J V 0/9?sz
0
‘ oo 00
=  q @,{9<+y)/+fe_(«+ﬂd
G D 4
= _ (r)(+y) Do
a L
—_':i
= 6 10/003): 6/4171; 6/)  (1+3) .4! .3 “e Okete :We 30/ 10M Ya/E Ihmduqc~ Final ~ Page 9 10, A game is played using an urn that contains a number of blue and red balls. On each play of
the game, a ball is chosen at random from the urn, and then replaced along with two more
balls of the same color. You play three times, starting off with five balls of each color. Being careful to identify any conditional probabilities that arise in the course of your solution, find
the probability ofthe following events: a). All the balls are red.
5) The last two balls are red. Lew" Xi cit/mic W eva' O? chained/13 a
red ball on Me Hi, Way (as. /.2,3), 0) P2m3=ﬁ;'  i,_}_.§__
A ’11., 11 10
t _3.
46
’9) P§X3XLX4E= FEXglxzxf}. Pflexf} PM}
— 3; i .‘E. , .5. 10 pts L Final — Page 10 a ...
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