part11

part11 - P[U>0]=1—P[U<0]=1_FU(0)= P[U<5 =1 Phwa...

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Unformatted text preview: P[U>0]=1—P[U<0]=1_FU(0)=% P[U<5] =1 Phwa = [-gwa=Fu<a>—Fu<—;>=§-;=§ P[%<U<%1=Fv<é)-Fv<§>=§-§=$ ,. Phwza =1-P[Iw<21=Fu<z>-Fu<—2>=2 ——._—————————— 7r . _ 3.16 a) The cdf equals 1 at 7r/2, therefore FX <-—) = 1 = c (1 + $111 —) — 2c => 0 2 b) Fx($) ' 1 tab-t win NP! 3.17 P[0’ g R S 20] = P[U < R S 20] = Fn(2a) — FR(U) = e_% — 6‘2 P[R 30]=1- P[R S 3a]=1— FR[30] = 6‘9” 3.19 a) We use the fact that the pdf must integrate to one: 1 1 x2 x31 c 1=Afx(x)d@=c/o $(1—$)d$=c|:§—§]0—6 : l 3 3/4 3:2 $33“ bP{—<X<—}=6 1— = _—— =0.34375 ) 2— ‘4 1/2“ $sz 6[2 3L” c)Fora:<0,Fx($)=0;f0r17>1,FX($)=1 ForOSxSl Fx(:c) = fox fX(x’)d:c' = 3x2 — 23:3 C 3.27 a) P[{X S x}n{X > t}] P[X > t] { 0 x < t = F31 )—Fx(¢) 1fo0) x Z t > : { 0 2: < t 1_c—Az _ 1_e—At ‘TufigfiT-l 3” 2t 0 J: < t -—r—‘—A::i—M a: Z t F,(a:la: > t) is delayed version of F423) —)\x b) > t) = 1 51%)“) :: ti“ = )‘e-Mz-i), x at fu4X>a /\ [X>t+x|X>t] 2:20 PM > t+x}rT{X > t}] P[X > t] l—Fx(t+$) l—Fx(t) 1—(1—e"\(‘+’)) m e-Ax P[X > 2:] The probability of waiting additional x seconds doesn‘t depend on the previous wait- ing time t . It is the same as when one begins to wait. fpower(y) = ————h_ 0 y < —a 3.57 a) Fy($) = Fx(y) —a5 3; < a I 1 y 2 (1 PH?!) 1 Fx(a+) FX(— —J~;__) —a a y From sketch of Fy(y) we see that fY(y) = Ffly) = Fx(-a)5(y + a) + fx(y) + (1 - Fx(a))5(y - a) for lyl S a and fy(y) = 0 elsewhere. c) For y < —a, Fy(y) = 0, For —a S y < a: For —a£y _<_a: mm = [/_; mmm + a) + mg) + [/m imam — a) 3.59 a) ForySO P[YSy]=0 Fory>0 P[YSy]=P[eXSy]=P[XSIny]=Fx(lny) ‘_0 ySO 'fiW)‘{Eflmm y>0 Fory>0 f( -dF —Pa)im Yy) — dy v(y)— x nydy y 1 = gfxflny) b) If X is a Gaussian random variable, then 0 y S 0 My) = e-“w-mW ‘ >0 yma ” ...
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This note was uploaded on 11/29/2009 for the course EE 131A taught by Professor Lorenzelli during the Fall '08 term at UCLA.

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part11 - P[U>0]=1—P[U<0]=1_FU(0)= P[U<5 =1 Phwa...

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