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# f10s - Math 135A Winter 2010 FINAL EXAM NAME(print in...

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Unformatted text preview: Math 135A, Winter 2010. March 20, 2010. FINAL EXAM \ NAME(print in CAPITAL letters, ﬁrst name ﬁrst): K 6 ( N AME(sign): Instructions: Each of the 6 problems has equal worth. Read each question carefully and answer it in the space provided. YOU MUST SHOW ALL YOUR WORK TO RECEIVE FULL CREDIT. Calculators, books or notes are not allowed. Unless you are directed to do so, or it is required for further work, do not evaluate binomial symbols, powers, etc. to give the result as a decimal number. Make sure that you have a total of 7 pages (includingthis one) with 6 problems. llll C) TOTAL 2 1. Bob tosses a fair coin 10 times. He wins if all tosses are Heads or if all tosses are Tails. Otherwise, Alice wins. Note that 29 = 512, and assume that in every year they play the game once in each of 256 “play—days.” (3) Compute the probability that Bob wins in one instance of this game. :FCBOL WI‘Aﬁ3 _’ :10 : E45.— (b) Compute, using the relevant approximation, the probability that Bob wins exactly twice in a year. m «w M u EM (mi‘éﬂwmméé) Z 3 — 4/2» -4/ I — a L 9 if \’°< PCZ wu‘m) “Q... —— I t ‘_ 2., <F (0) Under the title Alice losing her touch, newspapers report that “although Bob did not win even once last year, he already won at least twice in the ﬁrst half of this year.” Approximately how likely is the statement in quotes to happen by chance before the games started? ?Cw WM: 959*" <jcar/ >2. wiw ‘HW auw> :F( w//—-—— 3 'F( ~”~ ) a: U" ' € 4. — [4/4 - 4454/4) 964/;(4‘E’5W) CM. 0: w'w m 1; WM 3”“ .4 ”A” l’““m€ ‘9') 3 2. Fifteen Scandinavians: 4 Swedes, 5 Norwegians and 6 Finns, are seated at random around a table. (a) Compute the probability that the group of Swedes ends up sitting together. 4|. «4! g 4%! .—————-———" ——-——‘—"""‘—_" (b) Compute the probability that Norwegians have exactly one non-Norwegian person between them (i.e., they don’t sit together, but all ﬁve of them are in 6 adjacent seats). gl‘l’: A ﬁkaﬂl Mmca/{otw .m,‘ )J .4.» [4D . 5! .' (‘35 —2':! Omar a Arrow/~64 m 5 ”HM at m' manna/{M WMWW“ (brim; rm aw ‘HA—LM . 0. NW, not W ‘HLL » 'Nor ‘m MPH»; cwdv .414. 4ol IRIS! m a M (0) Compute the expected number of Finns that end up siting next to at least one Finn. N: #04 :Fimng W much “he deﬁant- MLTFI'MM N: I4+“' +1; It: IiFan ': 9+5 Mx'l’ +0 «:Fl'ﬂwy —‘ .—— ) - ,i,_<?__§§ EIL' 4 4w43 '21—” 0‘ ' s4 EN= 6‘5" _,_,,, 4 3. A math class has three sections with 40, 35 and 30 students. A committee of six students is chosen (without replacement, of course) at random from these 105. (a) Compute the probability that all chosen students are from the same section. 40> +— < 35‘) #130) < c. “ 6 . mm W (5%?” (b) Compute the probability that a section is represented with three chosen students, another ‘ one with two chosen students, and another one with a single chosen student. (“35> (seem +(‘é°)(3f7{3: +(:°M:?<i°) * “5) ('3‘) (2°) +05% W") + ( r) ( seal ” (c) Compute the probability that all sections are represented. Ac: {#akM ,‘ .44 ,\ Mfrs-“03 1 f’ ?CA4UA1UA3) _ p = 4 — PCA.) ~PCAL) —?(A5)+?CA\CAL)+P(A.DA5)+?[A;QA\$) 4. R011 a fair die repeatedly. (a) Compute the expected number of rolls needed to get the ﬁrst 6. GemrlN/‘QC‘%>/ h eﬂao‘lui thlggf 1': Q = ’ (b) Compute the probability that the number on the ﬁrst r011 1s strictly larger than each number on subsequent three rolls. Co‘maU'HM M ”HA 44+ mu anal mtg lZaLJM’ I 1;[?+;)+( -—-> (-4) (i) >1 ‘ F( \«imugl\ PC lathnﬂl) P(---l+zin+vﬂ'u6) (0) Compute the expected number of rolls among ﬁrst 100 on which the number obtained 15 strictly larger than each number on subsequent three rolls. ([363 ”imaUCAM "(YTOk 400. (MM NA (Ll) L (d) Determine the probability mass function of (X, Y), where X is the number of 6’ 8 rolled in the first 10 rolls, and Y 1s the number on the 10 th roll. (Write a formula for PX — -i, Y— - j) ratherthanalargetable.) ( i) (A ___>”'4 (”>5 -2 Jqu/ F(><= L \l/=\j> q“ C:O...1O —— ”G ___>4o—L) "’4 . 1 3° .31) 3- . A...) 6 5. For a year, a casino offers the following free game to all of its “guests.” A bag contains 4 cards, with respective dollar amounts 0, 2, 4, and 8 written on them. Guests enter the casino one by one, and each independently grabs at random one of the cards from the bag and gets as many dollars as the number on the card. . ‘ (3.) Alice, Bob, Carl, and Diane are ﬁrst four guests. Compute the variance of their combined Winnings. ‘ ‘ X; wmvvpas “K NJ. (jam Ex= a— (0—week?) 2; 2 V0~C><)= ~Ho +‘++’1A +Ltr) (32:) : 2| _ it; = 71.; VMCXA‘tXrJr Xc+x1>) = Li'Va/TCX) {3:9, (b) Assume "that 350, 000 people will enter the casinoM during the year. Compute the expected amount of money m the casino will have to pay for this game. n ; 3m 00 O 5n-‘-Xl+~+xm M‘ESH': V131; :gFD/oomg, (c) Keep the assumption from (b). Compute, approximately, the probability that the casino will have to pay more than m + 3, 500 dollars. Then approximate the probability that the casino will have to pay more than m —- % ~3, 500 dollars. {>an > m +s,wo) =P( §“—“%— > g/mo > was “'23: L‘. 3900 “we /:'r§;’3.:; =?(%>/2) Z 6. A pair (X, Y) of random variables has the joint density ‘7 A C(1+\$), OSySwSI, 0, otherwise. > f(x,y) ={ a) Find the value of o. Throughout, use that 1x” dx: L, for” > _1_ 0 n+1 I y _L _L ‘— g /\= C SoleCH—x)ol3= CSCA+X)NdX=CCL+\$)—c'? C: O 0 ° Fig] so?) (b) Compute the marginal densities of X and Y. Are X and Y Independent? :f X Cx) = {:- SXOH) a) = é-X‘C'WIQ / X 6E9 U 0 W) H533 =52; stoma =.; (1-5 + gm ' 7’ d = 3:4 (2—15 + ““5? - =—:=C3'25 1% NEW 1:093) % new (‘5) a» «it may. ( o o'HLU‘wI‘w.) (c) Compute the conditional probability P(4Y — X g 0l|2Y — X S ‘0). | X x {/8 dx S lqc1+x)d5 S C"“"‘>'T‘:d’( o o =P(\(<1‘FX>= XL 5 Q PC \(9 '42: X) 2/82” SCH—9&3 §("“")‘ 2, ’lxc J o - . .1. 2.. (d) Compute E(X Y). \ l X %'\$dx§xu€1+><>dz 3 i4- _4—— =2. :v =‘E L[— b’ "3 9,0 400 ...
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f10s - Math 135A Winter 2010 FINAL EXAM NAME(print in...

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