This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 (incitewk"
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 11 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 ll1Uncountabe $2 andDistributing 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
Fiﬁx): ~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 lﬁy <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_ iii251: <03
1“ 5e". if‘O 5x “ . 173 5...—“mt—mut—g.ml—mImu..I_—Impu———Iﬂ——w—  Exercises at Evaluate POX] < 1)“ _
b.‘ Decompcse FX into its' discreteand 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): % 1f15y<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 ofFdi . ' 999'? ...
View
Full Document
 Spring '05
 HAAS
 Probability, Probability theory, Following, CDF, ......._l..., decompose fy

Click to edit the document details