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real final test

# real final test - Fall 2010 EL6303 Namew(Show all your...

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Unformatted text preview: Fall 2010 EL6303 Namew (Show all your steps for Problems 1-8) Final Exam 1. (10 points) Y: g(X). 2, for x<—1 1, for -1\$x<0 y=g(x)= —x+1, forOSxSl O, for x>1 . ., Find and draw and interms ofFX(x) and fx(x)_ fx (3‘) x Mg' V'gaC“) 0% Alum 6m cww‘) Egg ﬁg: Vég) 3' Mﬂ‘al) % X£471 g >2” FY(%) 3‘ CL xé'lg) .: 3Cyl“) ' I :2 _ ﬁll): *8“ “DC * (i) ’ I am) ’ rm: 1'” v , ~ 51 w ~ iééléz/ \I F FX(@,)Wle(HL) r a: (L 7 [:y Cal) )WAMQMﬂW)‘ maul ,Fﬂ‘é)’; “Lg ’: vﬁ’CX‘éW‘dl) I » . , , duﬂﬁva) ' 4; kayz PM): RH) ‘31:?» 2 \~— F“ (71) lf'll A f h (C?) ~ EH) '3“ 'L \/ v a CH) O 4”” o 340 _ h a '2’)! l _ DAréMHHDCJwD Um W QCUW mmle UW‘ ,mcl l &0 to (\e Fx C > 2. (10 points) Find the constants A, B such that Ae'x,——1stl Bxe‘x2,OSxSZ ﬁ(x)={ : f2(x)={ 0 , otherwise 0 , otherwise are probability density functions. 616M «Eat mmbi’mg (Emma W I! % Jﬁm-dx : 420. “twig: j¥I(x)‘0iY:ﬂr a 40 i ~>~x d“: (L 9 yo 6i i 41 rd, :5 j/xei-W ”’ LHQM 1W Wﬂwi -4} ‘ _ .V1 1 t i x“ . V E j) __ A (E, ) 1 \/t OM? : “II/Hf :) :: flairji ~0°3C787 ‘ Gig—C, “POT E) 99 Jwidx 1'" KL "co .3 i ‘1 2 CL ,> J B?C ﬂx‘dx ’ 4 t L O Kg 1: A w WC? : t a) :E/i 6" at i i @‘L&XZ?‘:} o 8 (CAC)A[ 2’— 4— :m ” o (Xax a: FE 3. (10 points) Box 1 contains 4 white and 4 red balls. Box 2 contains 5 white and 5 red balls. At random, we move one ball from Box 1 to Box 2. Then, at random, we pick up one ball from each box. Find the probability that both balls are white. 0+3 5W 6 , B *1 REF 1 ((1)51) Cgl) WW cote fhoo (MM, . 80x CL :90 gm 2 (D W mow M 41a“ fﬁm‘ at)“, 50 (ML {MHZ WAG (LU/Mile 1M {ho/MA (19% ital/JR) ‘ V New? 1» WM WDC b0 a bow (haw/3qu .anLq ‘ -—:; i) U,ka WUUZ XLOLM/ggm ‘ B [L d0 W “ﬁg mo ) L : i549 “WW” ‘ é? ’ 02‘ pww 3 xi, (Li; 1: w P<y§§i : “47‘ OMOA (Newt) 7 %H J50 {9(5on mm m Judd/.9) / 4. (10 points) fX,(x,y)=1/7z', x2+y25k zero otherwise. Find and draw f(le=0.5) and E(Y|X=~0.5). 1% {PM <9”) : / wag“): WWW "‘9 Hm ‘o&__ H WM) “a “1:5 5 M ‘X 0 X( ’ o? r 4d OWMM “in 4662105) N” J3 \3 omﬂwu o ‘5” 5‘7 a?!) ab r H/ 3 g H who? a m X: MO AD {5/2 d) : f) r: j Xi \g 2 'B/ 5. (10 points) Y has a distribution with parameter A, y' ._ P(Y=yi)=l 31f! 1. Find the ML estimation of 1 when ‘ I'- y1=1.2, y2=1.4, y3=1.3, andy4=2.0. : ﬁ 01:1 Lg} A” F” L 4 j ' mm AA ‘ZKJL » Ar mm 3 Z <6 0 ~ Liz/x53» 4 ##ﬂ V ‘31 1 ‘31 ‘42qu L15 ‘ q 9 A b b ‘ x I “f0 (pk ML gmwrm AA (kw/CHM Mix‘f A mem Q’m Moo 2} ’ 3; 'i “M fa; M E ‘6 r -“47 j ‘ W LIT/1 d» ﬂ Li m L031 H and WM? [93 \fi/ . W b :> 0X 1W Wh‘éﬂﬁﬂ‘) 50 Th 086k £5 4 6 cm 117:??1 Am «4” /3;’;‘% :) 33W /‘ y 6’ ~ 3/ , w 4r x“ 4 a (ii W / 6. (10 points) FindA and B so that AX 2+3 is the MS estimate of Y. : E ’ f 2— 3> " EO/XL) _. A EUVW’BEMJ :> : E<V>®"'B EM) -/ [9 EM“) 195% 63 (Md @ A A ;..E “W *(EW‘A HWEM) W) ‘ ‘ '3) A- W) -~ Ema- mammal) (27> E: E(Y)~(E§_XL)'E(YXL)H5005(va E ‘1 ‘ ECXU~§~(Y9 EM W > ‘ EWEWXZ ' .. ,7 ' “3/ a Z ’ {5994 MT" L JFME/wggxa 7. (10 points) X and Y are independent with exponential densities f X05) = ae‘“‘u(x)= f ,0’) = ﬂe‘ﬂyuO’), ﬂ 9t 20% Find the density and distribution on=X+2Y. I .— ﬁv ‘ , .w- r 3* g¢2< 5M7 Jew): “gﬂmb My)” [36: a!) /. 2—;713’” Q I Akﬁ’ yaw / M W W M” / b i CL WU X&Ld0w VOdM’CLb14 tﬁﬂ %s¢%/ %w>:9‘wﬁ%‘3 {[email protected]>:» ;ﬁg:f§2, : i} E C U(,@) 3%») _; 6‘" Mg 005 M® 1-: ><+2Y c X039 W W Mokfwdkgfwﬁxaxww 33 Hm’ YaolaW/l’ 8. (10 points) The joint density ﬁmction is given as fxy(x,y)=Ce‘xe‘y, OSny<oo; zero otherwise Find C, f X(x), fy(y) and P(x+ySl). AreX and Y independent? X and Y are uncorrelated? Prove or disprove. 9. (Multiple choice questions. ONE point each. Explanations are not needed.) V 1) Which random variable has the second greatest variance? fix)- fz(x) 1200 Y x Vg 1/2 3 ﬂ, 4 i A x‘ PM ' C5) ’ -1 (2) 1 /bA —l (3) 1 p . 1200 ’5/9’ (3/2»:2 , (4) l/ .. 2) IfX, Yare marginally normal, the v&/ (6) X Y must be jointly normal. (a) True. VCbiFalse. ® {/J 3) In Lecture 11, "6 is a consistent estimate of 6?" is deﬁned as n%9=9—— @ k/Qa)’ in mean square sense (b) in probability (c) almost everywhere I (d) in distribution (e) None of above \/4) If X Y are independent and normal, thean Y must be jointly normal. M True. (b) False. \/ 5) If X,, Y are jointly normal, then X; Y must be marginally normal. ﬁr)” True. (b) False. @ \/ 6) ‘ “ “Independent” always implies “orthogonal”. (a) True. xﬂo’)’ False. Cb) V n The deﬁnition of “uncorrelated” is (a) P(XSx,Y.<_ y)=P(XSx)P(Ys y) (b) E{XY}=0 ME{XY}=E{X}E{Y} (d) E{X}-E{Y.}=0 (e) None cfabove I V 8) .If X is orthogonal to Y, andXI Y are jointly normal with 0 mean, then X: Y are independent. M1116. (b) False. @ V/s) If X and Y are independent and exponentially distributed, then X+Y must be exponentially distributed. ® (a) True. Wake. if '10) IfX and Y are jointly Poisson and independent, then X+Y must be Poisson. \La’)” T rue. (b) False. ® l/r'11) For arbitrary random Variables X and Y, if rx, = 0 , thenX and Y are independent. (21) True. MFalse. ® V12) In general, E{E{X [Y}} is a____. "(a) function of x (b) function of y (0) function of X 9 (d) function of Y (2)” constant (d) None of above ”; l/ 13) HA and B are independent and P(A) at O,P(B) ¢ 0,2 L It a then A and" B can’t be mutually exclusive. (3L7 M True. (b) False. ‘ V” 14) If P(A l B) = P(A), then We In have P(B IA) = P(B) (a) True. (b) False. ' view) If P(A I B) = P(B), then we (a) True; vflﬁ False. \f/ 16) If P(A I B) = P(B), then we ‘(a) True. {5) False. 17) A probability d ity function an be an odd function. (a) True. False. b v”) 18) A probability density function csébe an even function. f \,(«a.’)’ True. (b) False. {/119) X and Y have the joint density function fﬂ(x,y)=1/2, O<x<2, 0<y<1; 0,0therwise. Then, nylx) equals (a) 2y, for 0 s y S l; 0, otherwise (0) 2, for O _<. y 51/2; 0, otherwise (e None of the above. 7 V”! 20) In)general, E{X | y} is a__. Q) (a) function of x V615) function of y (c) function of X (d) function of Y (e) constant (d) None of above must have P(A (3 st have P(B ] A) = P(B). Ln=mn V09) 1, for 05y \$1; 0,0therwise (d) 1/2, for 05y \$2; 0,0therwise ...
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• Spring '10
• CHEN
• probability density function, joint density function, probability density functions, QCUW mmle UW‘, LHQM 1W Wﬂwi, WW cote fhoo

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