part26

part26 - 4. (15 points) A Gaussian random variable X has a...

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Unformatted text preview: 4. (15 points) A Gaussian random variable X has a probability density given by 1 x 2 fx(x)= e—HT), —°°<x<oo. I’m/27!: Find the probabilities below. Express your answer in terms of the (I)(-) function, given as I 1 _12 ¢(x)=[ me "hit. (a) (3 points) P(X g 3). Answer: @(0) __ l/ " l. Fxb) : 7' {/2 e H?) = w.) (c) (3 points) P(X 51). Answer: @ G g - 3 a c l : 450%) 3 $0134): IHWXLCI) =1—i(‘*«'§)= Paula) (e) (3 points) P(1 EX g 4). Answer: _ = EM— filll = man—ML?) : eel—16.0%) 1. (45 points) The joint pdf (that is, the joint probability density function) of two random variables X and Y is defined as k OSISLOSySL ny(:v,y): 2k1<x§2,1<y$2, 0 otherwise (a) (5 points) Find k. Magi onu/e jig prrtxj) 4.7017 ‘ I 10¢) + lizk) "— I (b) (5 points) Find Pr[XSl y<1] : S S Emmi?) AWL? - ‘90 a g 2 LXYL’quixa/y c: k Cl) 1 .L 3 on; 9.9 <6) www- w = j, amen ' i: ’ Till O I aMLUtfiL (d) (5 points) Find Pr[Y £1|X £1]_ -: ? { ‘I’ £1, X l: } 9”[Xe I] (f) (15 points) Find Pr[X g Y2]. {You may want to work this problem last.) +S§ZKOLEOUC ‘JR ' z (is) So mob: + éwmab: a 3' 371 I z @Hwé‘x )c, + 2(2 {Lg/1);, é<i3) 4 é—(wgm—mg) PJT4%+%+%@ '2. (45 points) The joint pdf (probability density function) of two random variables X and Y is' defined as 093?! US$31, OSySL fXY(13y) = { 0 otherwise, Where c is a constant. (a) (5 points) Find 0. Using The Norma/I‘eafio/L properfa ojf fhe pd! .' m " - \ \ QR Sgfly (N6) '2 j. S\(_ xa ._. i l=1 42> L9=i<=2 <2: 0 l — (d) (10 points) Find the marginal densities fX(m) and fy(y). ,yx (X3: Sou LLYUICP 056‘; R;Cxad$: Mxéz— (50 gm): gm oéxsi O eLSe wk ere 7M Pfobiem 73 50 déjmméfflhml flak we Can quer: 1% WE :3 “23$! ewe/mere” (f) (10 poifits) Find ‘ P(([X - Y) 5 0.5) u (X s 0.5)). IX-‘ll s Os (:7 43.53 X-Yso.s<:> —o.5+x 5% 0.51% @ ‘ hie/3(3) P(§}X-YI£O.53U éxéo'SEB {\-—;<:o;;‘-— - _ ‘ i x-OS : _ ‘1 ' \z _ 1 ‘1 :éEL‘Xadadj; iijxé ‘0 ax i”3”" : i—QSO’U ~xZ+O-QSx)dX = bit-1:224)» <$—:,+;)32@ X)“ _ fiemm => NOR fiaffilml prdb/LA N11 so”; 719 Check 0 1) “fnfiong/e X hé’IaW’ bflawe jtfiflxpd) A nof confioni‘ , ‘2. (15 points) X and Y are two random variables. Define two new random variables U and V by U = max( X. Y) and V : min(X, Y)‘ (a) (5 points) Find the marginal cdf of U. that is F1;('lt), in terms of FXy, the joint cdf of X and Y Note that you?“ answer should be t’IIT(Jlll.Sl’ITHl‘!/‘ in. terms of the function. FX 3,‘ and u. iUfWEZEWCX-‘lleuf : {golkXMOll/we Eu? mm): mm) = (b) (10 points) Find the nmrginal cdf of V. that is Fit-(v), in terms of ny, the joint cdf of X and Y. Note that your answer should be exclusively in terms of the function ny and v. Eve-v} i maimed = €><°r Y ;5 av? ) O Y: of (x-— T + 1)) (x1 ? (X: ): ? ._\l/ + Cx= ‘? )2 LL 5’ Va [/3 + + Vb ?(U70) wLare U :5 an GamSS;a~\ ("Wm VNI‘ALQ wi‘Hsmm”7 MA dwfmce 1? 6‘43 E 92.) uJL-I-re E (‘3 Gauss-'CMIM:OI Eli/a 1 611.? :— 9v; 1.; [pm-1)) = [F(‘|4/\/ £|)+ N/W’fl // POI/>0) fl firm) L x=—.|m)= f( ) FLY70) : N7) .l/Z m 1:7 ‘ k X”\ l9 NW'LL; ...
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part26 - 4. (15 points) A Gaussian random variable X has a...

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