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# sample-m2-09-key - Stat 126 Exam 2 Spring 2006 \A U S...

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Unformatted text preview: Stat 126 Exam 2 Spring 2006 \A U S DZLL‘LlhO'Vi 5 Name : ID#: I pledge that I have neither given nor received unauthorized aid on this exam. Signature: (1) Check “true” or “false”. (3.) If X N Uniform (~3a, 3a), Y N Uniform (~a,e)-, then VarX = 3 VdrY. I true I ; false. / (b) If X N Poisson (1), then 2X ~ Poisson true ' ;felse V‘ (e) Let X denote the total number of heeds among 10 tosses of a. coin (p), and X and 10 — X have the same distribution. Then p = 1/2. _ true ‘/ ; false (d) If the indicators I A and I B satisfy I AI 3 m 0, then the events A and B are mutually exclusive. \/ true ; false (e) Let be the total number of tosses of a. coin [with P(H = 3 /4] right before the 4th tail appears. Then EX = 15. I ' true ' I ;faise (f) If f is the density of X, then f(:r) = P(X = \/ true ; false ~ (g) If X and Y follow geometric distributions with EX > EY, then Va'rX > VarY. true #WW ; false (h) If X ~ Poisson (3), Y N Poisson (1), and X and Y are independent, then X — Y ~ Poisson I true ; false ‘/ (i) If both X and Y follow exponential distributions with EX > EY, then P(X > 1)'> P(Y > 1). true ‘/ ; false (2) Assume X and Y are independent Poisson random variables with means 2 and 3 respec— tively. (a) Calculate Var(X—2Y):: ("2.)1 UM [f 2 2 + I a 22 WW, X474) a T w (b)FindP(X>Y[X+Y:3):—‘ —~ -——:3——-——- :2 r—-" f9(x+y:3) 5 125“ K‘l’L/N (179955914 [5”) Z. ‘ gdxfyﬁﬁ‘fglw 3 _‘ r ‘ a (9;\$52 clwwmm = e % ' WWahW = F(X53, 7:0)+P(X33y:l )5FaﬁjfﬂJ'W-l’FH‘AZJPa/H) . :- €223 ég 3b '1'. o-Zzz #3 I “5., LL "5T '6? E —T e .1 :2 e ( 5 + J (c) Calculate EHX + Y) (X — ' . 2’ ' l : ECXE—Y"): (Ev—"EYE": VMX+@X)1” “W7"@7’)l (3) Two players A and B roll two fair dice independently and simultaneously at each step. The game v'vill end as soon as the two players get different digits among {1,2, 3, 4, 5, 6}; and whoever with the greater digit will win the game. (a) Calculate the expected number if steps needed for the game to end. 1“ mater , ’F’the): at“); / 7’0” ﬁt): :2: - tej‘ K=#D£S+&P5%£2Amw,m mkﬁpé -' 1 (5K :: "‘ = 75'? . (b) What is the proiability that A wine the whole game? 7% {3; wins t: "E wl'us M 9m {:4 h 'S'ILeFS) ox) ’9 . .- ' : r-Z‘ (F(+,\¢9 F“ M I“: n”! steps) , Wms {‘14 7‘50). 30%: . w nvt Li.“ 5- ; _ ' w anfiusppaggxus} W :3 'Z . w 1:. Xh—g h (242'. —'“ 2f :(IAWMS)+F(B.WIH5)+P[ .3 17 ' L :J'. In t , :57 504mg) 2. L/ (4) Let X represent the total number of four events A,B, O,D that occur. Assume P(A) = P(B) = 1/2, 13(0) = P(D) = 1/4. ‘ (a) Calculate EX. 5: + )7: ‘3 F—‘M H 1 (b) Assume events A, B, O, and D are pairwise independent. Calculate VarX. Va} (114+ 13 + IC + :3) J : vat, + UM%+ VMIC + vat/r13) : (mmﬁwm) + WE)(I~FCB))+ WM? Pro) +— PKD)[/~Pzi>)) \— + I...w 4. «Fl—— {IN -_.a tr!"- J‘. a a tamtz-L \$9 ._ 'l y—q 8 ‘ . C: 0.875”) ' (5) Suppose X has the density HI 3 as S 1, otherwise, Where c is a positive constant. (3.) Find the value of c. , l _ 1‘: 1 =59 C : 7Mr—H : \$91" If C 79 ‘ f xzdx —- ~i (b) Calculate PUX} = 1/2): (9 \\ FCX=%)+F(X ti) / I l (c) Calcuiate FOX} > 1/2)"; 2 it 'Xz‘cl ‘x ".2 36'"5/ g; i . ' I 3 / . _, z _ (d)FdeX.._. «)4. Ex (1% _ 1 ix? 0/76 a} W M f cm W 151 6 Find Va'rX. = 1’» ,__, ' U EX '— 061' “3962356 zf/s “I ’2”. 5' f "I ._ ,3. l. f H '3 ‘"‘ 7- 5” ~1 "— '5; (6) Suppose the numberof jobs sent to e éomputer server follows a Poisson process with the rate A = 3 (per hour). Note that it is equivalent to assume that the interarrival times-of those jobs are iid exponential random variables with mean 20 minutes. (3.) Given that 6 'obs arrive in the ﬁrst hour, what is the conditional probability that at least 2 jobs will arrive in the second our. u " W P, Mole 7935M“ W491i; A M8 5 Me le-Z 39033 A): HS) :2 I ~ F(X=o)~z°(x=/ ) (b) Starting from 9:00 am, what is the probability that the 3rd job is received by-the server before 9:02 am? . ‘ I V“, ’Fol‘gg‘m-(ﬁ) 51‘”,ch A23 7.5/23) = 2-r(y:a>~r(y=o~ {DO/=2) C /u‘8—-Dl—l 0.! Fa é-D [{Q‘Uf a no.) ' my: 4w {robs were” 5“ W ‘W W m X'N?OI%W (A) . with i=3 _ my: xvmwwfmr we ~ 9. '3 we 5 ' re 0., WW ' ...
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## This note was uploaded on 11/17/2011 for the course STOR 435 taught by Professor Staff during the Fall '08 term at UNC.

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sample-m2-09-key - Stat 126 Exam 2 Spring 2006 \A U S...

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