Chapter 05 Problems - 170 Chapter 5 _ Describing...

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Unformatted text preview: 170 Chapter 5 _ Describing Probability II: Uncountable £2 and Distributions reptezsemation to? ‘_ I _ _ continuous. and:singu1at)thggt,tvillptn I ‘ ‘ _We closed with an cxainbi'e "911a (incite-wk" statistics ltfestimate {(305} 53‘ . id!" when he Lime ed! '; 7 - EXERCISES E5.1 By considering the elementary properties given by Eqs. (5.2), (5.4), and (5.6) that char- acterize a cdf', detetmine which of the following are cdfs for S2 m R (specify the property that fails when the function is not a cdf): a... ex/(I + 3’) b. UOC) + I1 — U(x)](1+ €x)/2 c.» 8—le d. Oif‘xi<0and1ifx 20 6.. 0 ifx 5 O and 1 fix a 0.. I E5.2 Ihe number K;- of ions emitted from a source in time T has a cdf' F]; such that ' FX(1)—'FK(1/2) = “1" 7 “a. What is P(K1-= 1)? ' b. What is FKG)? E523 Evaluate and sketch the cdf F(x) for the exponential pdf‘ given by fix) = Wm'axUOC), when 52 = 1R. E5.4 Use the exponential to evaluate the probabilities of the following events: a. {w : a) 5 x/a}; . b. Ufloko : 2k/a < a) 5 (2k +1)/a}.._ E5..5 Evaluate and sketch the cdf1F(x) for the Laplacian pdf given by t _. E "aim ' ' fix) 2e , when 9 = R. ' Exercises ' l 1 7 1 E55 E53 E53 _..___......._-l.............__.__—. mu_.——H_. mm.-—_mH—_nmu.-m————_ GDP for Use in Exercise .-...'——1___w..__ru___-r. o ‘ .... ..-;._‘....;l_.__.;_;_.._.._;.. —4 ---3 —2 —~-1 1 2 3 4 Discrete 00!: Example -——r—.'---——-r——--—-p——r -~3 -2 1-1 0 .1 Figure 5.5 CDFs F and G for Problem Use the Laplacian to evaluate the probabilities of” the following events: a‘. {wzw sac/a}; bl. Ugiffiw : 2k/a < co 5 (2k +-1)/a}‘. If tho cdf‘ 1. F1107): ‘1-+ e_x. evaluate P(0 5 X 5 I)“ 7 . Consider the two different cdf's F and G shown in Figure 5.6 fora random vmiable X .‘ For each of them, evaluate the following probabilities; P(X < -—3), P(X 5 --l), P(X =0), P(X > 2). 172 Chapter 5 Describing Probability ll1Uncountab|e $2 and-Distributing E53 a Determine and accurately sketch the cdf F for the 724 pmf when 52 = IR. - . b.‘ Using F, evaluate P ({a) :_ ~-..5 < to < ‘.5}).. _ £5.10 Determine and accurately sketch the cdf' F for the binomial 3(3, 1)“ E5.11 If ' fixaz 12 l Fifi-x): ~4- if‘05x<2, 0 otherwise then determine POX | < 1)“ E512 a If the cdf 0 if‘y < -»2 .l(y+2) if‘--25y <1 Fy(y)= “5 if l-fiy <2 , .25): if 2 5 y < 4 1 ‘if4'5y ‘ evaluate P(Y 5 —1), P(Y = 1),P(Y = 2)” (It may help to sketch Fy‘.) ‘ b.. Decompose Fy into a discrete cdf F4 and a continuous odf' Em, and determine the coefficient A. for Fd‘ ' ES.13 Detexmine and accurately sketch, for x _<_ 3, the cdf F for the Poisson with A = 1‘. £5.14 A sequence of 10 unlinked (independent) bytes has been received“ It is known that the probability is .3 that a first symbol 'in a byte is a 0, Let K be the number of received bytes having a 0 as first symbol. a. What is P(K = 2)? b‘. -What is F'KG) for the self Fg? ESJS The cdf (it may help to sketch it) of interest is 0 if}: {0 ' Fm) = 23:2 ifng <14. 1 _ otherwise an Detetmine P<X 5 —1),P(X 51/2),P(iX|> l/2),P(X =1). by Decompose the cdf'Fx into a discrete cdde and a continuous cdfF,m and determine the coefficient A for Fa" ' 125.16 The cdf‘ of interest is _0 if x < ~2- Fkbi): ‘.3_ iii-251: <03 1“ 5e". if‘O 5x “ . 173 5-...—“mt—mut—g.ml—m-Imu..I_—Imp-u——-—Ifl——w— - Exercises at Evaluate POX] < 1)“ _ b.‘ Decompcse FX into its' discrete-and continuous pans, Fu- and Fae, and determine the Coefficient l for F4” £5.17 Iffx(x) = x’2U(x 1), determine and sketch the cdf F'x” 1315.18 We are given that ' e? if y < —-1 FY07): % 1f-15y<1 1~—e y iflfly a. EvaIuaIe P(Y = 1), P(Y s; a, and [’0’ = 0).. _ b‘. Decompose FY into its discrete and continuous pan F4 and Fm, and determine the coefficient}. for Fa” ct. Evaluate the density ‘f‘ for FM. E5.19 The cdf' a. Evaluate P(X > «4), P(X =11. _ ' - h. Decompose F}: into its discrete and continuous parts Pd and Fm, and determine the mefficient A for 173.. c.. Evaluate the densityf for Fact. E520- Consider the cdf‘ Fx(x) = l 1 a.‘ Evaluate the following: - P(X $0), P(X > I)» b.. Decompose F'x into its discrete and continuous parts F}; and Fag, and determine the coefficient A for Fd. ' 174 Chapter 5 __Describing Probability it: Uncountable $2 and Distributions E521 The cdf (it will help to' sketch it) 31:9" if}: < 0 1 fi0$x<1 ,Fxcx) =' g _ _ ' i w' fi15x<2 3 1—lflfi fi2<x 4 .. - Eiralualc P(X s 1) and PG! 2 2).. Evaluate P(% < X 5 g)” Evaluate P(X = 0) and P(X = I). Decompose the cdf F3 into its disarete and continuous pans Far and Fm, and specify the coefficient A of-Fdi . ' 999'?- ...
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Chapter 05 Problems - 170 Chapter 5 _ Describing...

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