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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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- Spring '05
- HAAS
- Probability, Probability theory, Following, CDF, ......._-l..., decompose fy