midterm1solution

# midterm1solution - AMS 310: Survey of Probability and...

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Unformatted text preview: AMS 310: Survey of Probability and Statistics MIDTERM EXAM I Spring 2009 Last Name: _—_ First Name: _ ID: Show all your work for full credit 1. Given aset of observations: 12 30 3O 27 3O 39 18 27 48 24 18 (a) (5pts) Find the 90th percentile. £3, :8, 13; at}, 27, \$1.50, at, 50, 3‘), 48 1am, 19:01} {mp}: l0 (loll/x nevunh‘lt 7* Xua) 3 (b) (Spts) Find the minimum, maximum, range, and the interquartile range. min:— I}; mew: 4-8, Meiji ‘5 48415796 51:; Minus—ﬂ): X0) = it (915:- Xo‘n—ma—i) ; Xe) =3“ lawn-Mme may): : (9r (Ebr- ILA (c) (5pts) Draw a bOXp 0t. de= Win-091) =’— We = 37 l3. 1% 37 3‘0 2. Let X be a discrete random variable taking values in {—3,0,3,6} with probability mass function P(X=—3) = %,P(X=O) = §,P{X =3):}1,P(X:6): g (a) (6pts) Compute the mean E(X), variance Var(X), and standard deviation of X. 12131:”??? POW 0‘) = “d‘ir'f‘r 015+ 3¢+ 5J5 2: I ECXZ); 'i" 3 %L r 1. ’p-ﬂ_ 9 'c. at!» a, ‘1‘”. \r = a Var (X): ED; 3’ m1 «- r I 2‘ Lg— )tendetd bliwédmfh J}? :33, g (b) (6pts) Calculate the probabilities P(X S *2), P(X 3 3), and P(X S 6.2). Vase): P(><='5)‘7-Lr ; poaé 3) poo: «3) ~+Pfx=o) +th =3) wig PCX-ééa) =1 3. Suppose that P(E) = 0.6. What can you say about P(ElF) under the following cases: (a) (5pts) E and F are mutually exclusive? pt’ElF) = 0 (b) (Spts) E and F are independent? mm = Ft?) 3 “‘5 (c) (Bpts) E C F? . _ pct/IF)“ (E M PiElﬂ-“ﬁﬁ ' - pa?) =) M, spams: \ (d) (5pts) F c: E? ) -FLEQE W) FW‘T’)’ Fm PW) 4. (8pts) An airline company knows that 5 percent of the people buying tickets on a certain ﬂight will not show up. The company has just sold 52 tickets for a ﬂight that can hold only 50 passengers. What is the probability that there will be a seat available for every passenger who shows up? ire/l” X“: tibi— paéswﬁw twins 9th Mr ‘x N twin, Mb“) P( X550) 2: i, F( x=b‘lv)’ P(){»=5’a) 52 2 ’_ ('91) cﬁgﬂruc; , cqg’ : I» yaw-ﬁﬂwew «35‘ 5. A basketball player can make a free throw with a constant success probability p : 0.25. Let X be number of free throws needed until his ﬁrst shot. (a) (ths) What is the distribution of X ? Write down its probability mass function and ﬁnd its mean ”' X M drew (my) .- ,1... 1 . past): 9.154%" 0.75 Em! M. 4 (b) (Bonus 5pts) Let Y be the number of free throws he must attempt in order to make exactly four shots. Find the p.m.f. of Y. .- w — - A ‘3 i p- :5 ‘ "_ [VJ ﬁve): (21;) Ms“ 075 “lb .. , ,Lq- 6. Consider a test for detecting pipe corrosion. The test has probability 0.7 of detecting a corrosion defect when it is present, but it also has probability 0.2 of falsely indicating corrosion. Suppose the probability that the pipe has corrosion is 0.1. (a) (513%) What is the probability that the pipe has corrosion given that the test indicates its presence? Lop E2 : i Hm fey” Indians corrosion} A 9 ‘H/UL F? [24: Wu) warm/314m 3 PLBIAVM Helm-46+ Pcmw-l n6 ‘ '(A a 07x64 mm; M - Pm) pus:n;yu+>+p(51/+)pm) 3,.7xa;+o.uo , " a? WWW (b) (5pts) Find the probability that the pipe has g“’gorrosion given that the test is negative. I) vw;nA.@_E:2_i < mu” M .. FMIB) FIE) NEWPMHFWmm) 0.2,”: , ., . ; a 5 0;.5 xvi; + 3.3 x at? 7. X is a random variable taking values 72, *1, 1, 3 With probability mass function 1 1 1 3 P = i = 7 2 — z — = : — : = 7 (X 2) 4, P(X 1) 8, P(X 1) 41 P(X 3) 8 . Deﬁne a random variable Y as Y 2 X 2. (a) (2pts) What are the possible values Y can take? g i , 4-, ‘l 75 (b) (3pts) Write down the probability mass function of Y; gym/=3): P( XWIH P(X==l): “337’ P(Y37) 3 WW3.) WWW" Pariah—E; “"13” (c) (3pts) Calculate the mean and variance of Y. rm; :v%+¢~i;+ ’2’ “"75 in, «~- _’3‘7 ELYl-W g+ro-—g;+s|1§—~, 1 w 8. (10pts) Consider independent ﬂips of a coin having probability 0.5 of landing heads. Say a changeover occurs Whenever an outcome differs from the one preceding it. For example, if the results of the ﬂips are H H T H T H H T, then there are a total of ﬁve changeovers. What is the probability that there are exactly I: changeovers in 71 consecutive ﬂips of a coin. Y: lwirvl ﬁr» oiwﬁmws . , J, A hﬁi H cfr LhmﬁeﬁwrS [n Tl CwQewa/i Snys '3 YI—I @5th dwygwsr OC’JLLWS Luin Wbcrlagmg a 'I .a I2. , -7 prx=m= (T) ti“) Hg)” ’= (“h'lIJgih‘ it! 9. A man has ﬁve coins, two of which are doubleheaded, one is double-tailed, and two are normal fair coins. He shuts his eyes, picks a coin at random, and tosses it. (a) {5pts} What is the probability that the lower face of the coin is a head? LU” H, a; H“ dew 25, ct lac-d 7; {3‘ ’5' (X deﬂate -i/Vu4.cl-ul be {In {S Picked} gr. ; a some. were“! “in is Frauds 6,53 Ci hwmal coin is 2 ,3 3— W4): PiHl/i)gri¥3+ erm‘I' 031+ PiHleﬁc) : e I ’5" ‘l 0 "L 3i '? ‘15" I (b) (5pts) He opens his e es and sees that the corn is showmg he what is the probability that th | lower face is a head? I I '0: 3 We. can. Cs sham/ab _melg mm: WNW) a. WLWWW . [7(9) " meAimiwmnwrwpuns)ch '3- 1 "5’ M : ; r: __ :2. 2.3... ‘ Jan}. a}; *” *"“I' I s—tﬂi’a 5. 5 33> (0) (Bonus 5pts) He shuts his eyes and tosses the coin again. What is the probability that the lower face is a head? LQ/i’ vexewwv 1(455 Cu PUEID‘) 2 “En” = FWD) CU) ' I : pigeDIMPUiI-I-PtEnplﬁ)‘(BHP{EnDIc) 0:) j. _:I. , _ 94L 1 I‘? “T 0+ 4”? 2—. : -—-———-:—~———- = 'é 5‘ 4 ...
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## This note was uploaded on 03/21/2010 for the course AMS 310 taught by Professor Mendell during the Fall '08 term at SUNY Stony Brook.

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midterm1solution - AMS 310: Survey of Probability and...

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