chapter 3 (1)

Prévia do material em texto

C I IJ\PTI;R J 
Sc:t:tio.n 3· 1 
3·1 ThcrJC1i:cofX is {o.l.2 ..... 1000f 
3-3 ThcrJC1i:cofX is {0.1.2 .. . 99999} 
3-S. Tho rooac ol' X i> {I. 2 ..... 491 }. llccolu>C 4!)0 JlWU ""'canlbnning. o. ncnoonlonnin!!- pori m1.t<t be ""locto;l m 
~~ SCl«:tl<liL~. 
J . 7 The r~c of X iHlOo\'<:rtiently nlOdclod os nll r~armesori'-.: itl(e~m. Thnr is.. tbe mn~te of X is {0.1.2 ... .} 
3-9. Thtl\ll1liCOfX i~ (0.1.'2, ... ,15} 
3·11. Thc~c oi'X i> {0.1. 2 ..... 10000) 
3-13. Thcl\lllScofX is {0.1.2.. ... -10000} 
Section 3·2 
3·15. All probabilitid 011\!' grc:ttr r than or equal to ttm tsntl sum to on.:. 
al l'(X S2)- I/8 +218 -2J8 t- 218 + 1/8 • 1 
b) P(X >- 2) = 218 + 2/S • 2.1'8- 118 = 111! 
c) P(-ISXS'I)-218 • 2/R nn;•6•R• 3f4 
d) I'(X S - l or X - 2)- II'S- :!ll~ + 118 · 418 · lf2 
\-17. Pm~nbilities on: ncnl1cl!"'h'C nnd "'"" to IJC1C, 
~~ r(X - 4)- 9/2$ 
b) I'(X S I) - 1/2S + 3!2S • 4f.!S 
c) 1'(2 s X< 4) = Y2~ + 112S = 1212.5 
d) l'(X "·IO) I 
3·19. X- numbcrofsu('{'c>siulsu~eries. 
P(X = 0) = 0.1 (0.3:1) = 0 033 
I'(X - I)-0.9(0J3) + 0.1(0.67)-0.364 
I'(X - 1}-0.9(0.671· 0.603 
... :u. X - number ar w~t¢oo tlut p:~<;( 
P(X 0) (0.2)1 O.OOS 
P(X - I ) • 3(0.2)1(0.8) -IJ.()% 
P(X = 2) = 3(0.2)(0.81' = 0.384 
P<X - Jl (0.$)' o .. m 
.3·2l. P(X - SO nrillicn) - 0.5. I'~ X - 2S mlllioo) - 0.3. 1' (X - 10 million) - 0.2 
3·1 
.1·2~. I'(X IS null ion) - 0.6. 1'(X- S millkln) -OJ. I'(X- -o..s million)- 0.1 
~-27. X numb<r af C(>lllf)Otla\1< that ITlC'Ct spccif>:.>ti.'!n$ 
I'(X OJ (O.OS)(O.O'.!)(O.OI) 0.00001 
PI; X II tO 95)(0.02)(001) • (0 05)1098)(0.01) • (0.05)(0 02)(0.991 =000167 
PI;X 21 (09SI(0.98)(0.01) + (0 95)1002)(0.99) • (0.05)10.9'8)10.991 =- 0.07663 
1'\X 1) (095)(0.9R)I0.99J o.?l1<>9 
3·29. X wailing 1imc: (hoW\) 
PI;X I) 19 500 0038 
PIX ~~ Sl ~ 0 .102 
PIX ) ) StdOO 0 11'2 
I'\ X 4) 102 soo 0.2A}.I 
PIX S) s;,~ 0.174 
I'( X 6) 6MOO 0.124 
I'( X 71 -10 ·~ 008 
I'(X 31 Ul•<(K) ()016 
I\ X 9) 14•500 0.028 
I'(X 10) 111500 0.022 
P(X IS) 10\SOO 0.020 
0.038. .t I 
().11)2, T 2 
017'2. ,Y 3 
0.204. 
·' 
4 
0 174. t s 
f(.r) 0.114. Y- (1 
008(1. T 7 
0.036. t g 
0028. • 9 
0.(122. t 10 
0020. y 15 
1.)1. X Non-f-liiC'd ~~odl dtpch 
1'\X 2SSJ tiS IS • I\<H)on26 O.HO 
I'(X 2181 267126 0.003 
P\X J 111 329()·7726 0 426 
I\ X 2 I II l49n72(; 0 ~5 
I'(X llo7J (280 • 887)•7716 O.IS I 
1\X - 2171 36:7126 O.OOS 
0005. y ll7 
0.00\ t liM 
((tl 0045. 
y 231 
0.370. T 255 
0.1 S I, y 261 
0 426. .. 317 
3·2 
' ,, 
.\• "-'· 
J.J). 
3-37. 
u. :c <-2 
1/ 8 
- 2 $.t <-1 
3 / S - 1<.<< 0 
F(.<) -
518 osx< l 
718 l !>.t<2 
I 25.• 
~~ P(X s 1.25)- 7/8 
b) P(X :52 2) = I 
wb."'tc 
~) P(-1.1 < X s 1) - 718- I IN -J.I~ 
f.,( 2) - 1/ R 
fx(- 1)= 218 
fx<0> = 218 
fx(l)- 2/S 
l 1 (2) = 1/ 8 
d) P(X >0)- 1 - I'(X S0) - 1 -SIS·JJ~ 
0. x < O 
0.008. OS:r< l 
F(.<) 0.10·1. 1< .< < 2 
().~R, 2s;x< 3 
I . 3 ->.< 
/CO) = 0.1J = O.OIJS. 
/(1)- J(0.2:(0.2X0.8) - O.Cl% , 
/12> = 3(0.2XO.SX0.8) = 0.384, 
f<Jl = co.sy = o.s12. 
0, 
F(.T) = 0.2. 
o.s. 
I. 
,T <I() 
10 $.V<2$ 
25 <.r<SO 
50 s; x 
when: P(X SO milli:m} - 0..5, PIX 25 n>illion) 0.}. P{X - 10 million) 0.2 
1·39. Tile sum oflhc p<~>b:Jbilhld I> I :u>il all probotnlilk s ore £1"'4lct rlnn 01 <>qUll ur :..:nr: prnl:jl l) - O.S.j(3) - O.S 
a} P(X S 3) = I 
b) P(X S 2) •O.S 
~) P(l S X <:2) P(X I) 0.5 
d) I'(X > 2) - 1 - I~X S2) -0.S 
34 1, Tile sum of rbc proll<1bi l ir ie$ i~ I nnd Ill I rroNt>~ilk11 nn: ~rcr r~n or oqwl l(l?cro: 
prn(: 11-10) - 0.25. 1130) - 0.5. QSO) - 0.25 
ol P(X ~ $0) = I 
b) P(X :i 40) • 0.75 
c) P(·IO S X S W)- rrx - 50)• 0.25 
d) P(X < 0) - 0.2~ 
c) P(O s X< 10) - 0 
f) P(- 10 -.: X -.: I()) P O 
3-oB. 
f), ... < 26a 
F ) 0.2~. 266 $ .<< 2?1 (.Y : 
O.S4. 2il $ .• < 274 
I. 274 $.< 
when: P(X - 266 K) - 1).24, P(X - 271 K) - 1).30. P(X - 174 Kl - I)A6 
0, A< I.S 
005 I J, , ... . . .) .oo.<•,' 
F(x) = 0.30. 3 $ ·' < 4.S 
0.65. 4.5 $-• < S 
0.85, S$ :t< 7 
7 <:t 
wh<'l1: P(X= I .5) = O.OS, P(X = J) = 0.25. P{X =4.5)=0..>5. P(X=S) =0.20. P(X = 7) =O.IS 
Sa:tion ;w 
3-47. Mcnn nnd vnt'Pncc 
1' - F.(,\')- Oj'(O) 1/(1) 2/(2) + 3/(3) -! f(-1 ) 
= 0(0.2) + 1(0.2) - 2(0.1) .,. 3(0.2) + 4(().2) = 2 
1'(.1'1 = 01 f(0) - 1 ~ /(11 + 21 /!2)+ 31 /t3l+ 42 /<4>- s•: 
- 0(().2) 1(0.2) I 4(0.2) 9(0.2) t I (i(D.2) 21 - 2 
3-49. Dcrcnnio:: E()() nnd V(X) (Qr r.Jndom vro~lt in c:<crcisc 3·1 S 
Jl = £(,\') = - 2 {(- 2) - 1/(- 1)+ 0/(0) - 1/(1)- 2/(2) 
- 2(118) 1(2/ K) 0(218) t-1(2/R)+ 2(1/l\)-0 
I '(X) -2:/(-2)- 1~ j'(- 1)+0~ /(0) + 11 /( IH· 21 Jm-1'~ 
= 4(1 / 8) + 1(2/8) + 0(2 / 8)+ 1(218} + 4(118) - 01 = I.S 
.1.5 I. M~Jll Md •umncc liltc.(ereisc 3-1? 
1' = I:( X ) = 0/(0) - 1/ll) - 2j'(2) - 3/13)- 4J'(4) 
=0(0.1);) - 1(0.12) - 2(0.2) - 3(0.28J - -!(036) = 2.8 
1'(,\')- (11 /(0) I' {(I) 21 ((2) + J1 /(3} + 41 /(4) Jll 
- 0(0.04) ... 1(0.12) .,. ·1(0.2) • 9(0.28) - 16(0.36) 2.8' - 1.36 
3·53. Mc:m nnd vurimcc ~ r.on~m vuri:>blc in cxcrci.<c: 3-IIJ 
,, = E(X) - 0,{(0) - 1 ((1)- 2/(2) 
- 0(0.033) + 1(0.36-1) + 2(0.603) 
- 1.57 
I'( X) -Ol/(0) I ' /(I) • 21 {(2) 1o1 
- 0(0.033} "' I(OJ6<1) ·t4(0.603) - 1.57' 
= 0.3 111 
3·55. Oetcmoinc ·' wlwn: roU1£< i>(O. 1.2. 3. x);:md m..-.on IS 6. 
11 = £<X) = 6 = 0/lOi- 1/(1)- 2/(2)-'- 3/(3l+ 4(x) 
6 - ()(0.2) + 11(1.2) - 2(0.2) - 3(0.2) .,. .r(0.2) 
6= 1.2 +0.2.< 
4.K - 0.2.< 
X= 24 
3·Si. X - numbcrarcDmj)utcrs tloat vulc lor, kn roll wl:.:n o risht rull i> "''I"""P<i:r!c. 
I' E(X) o•tlO) + I• Qt) + 2•!l2) +3•Q:l)H•O ~l 
= o • 0.0003999 * z•s999•1o·• ~ 3•3.9996•1o·u + -£• t•ao·••= o.oo().: 
s 
VI X) = :[ f(x1).x1 
i I 
' 1•)• - 0 000399!16 
3·59. (a) 
(lo) 
Tr3u5o:rhon Fn:qucn<)' Sde-:1>: X 
Ncwardcr 43 ,. _, 
I' 
E(XJ = 
Pl•yment 4.£ 4,l 
Ordc1 status 4 II.~ 
llei i\'<IY s 130 
Stock IC\-cl 4 0 
TO(., I 100 
•0,.;3 + 4.2•0.# - IIA•O.().I + 1~"0.05 + 0<0.(); = 18 694 
s 
VfX) = :[f<c,X.<- ttl2 - ?JS.%4 u = ,(V(Xl = 27.1287 
••• 
T rnnsocaioo f'n:quc1~· All O(X'f'JtiC41: X 
(\X) 
0.43 
1).44 
0.04 
o.os 
0.0·1 
Nc~> otdcr 43 23 - 11 + 12 = 46 
l'.l)' ll)CRl 41 4.2 t 3 -t I - 0.6 - 8.8 
On.kr $1::t1Js 4 IIA I 0.6 - 12 
Oc:livcry s I~+ 12(1 + I() '21Yl 
~'klc\'CI 4 oq = 1 
Toanl 100 
11 = E(X) = 46•0.43+ 8.8"0.44 + 12"0.04 + 200•0.05 - t •O.().I = 31 .I 72 
' VfX) - L f(x, X.•- tl)~ = 2947.9% 
t I 
jp() 
OAJ 
0.4-1 
0.0•1 
1).05 
0.04 
-
'·61 . )J - 11(:\")- I (0.038) ' 2(11.102) 3(0.17!) I 4(0.204) I 5(0.174) < 6(0.124) I 7(0.1A';) I 8(0.036) I 9(0.028) 
• 10(0.022) " I 5(0.020) 
= 4-SI)S b:lln 
s 
V(X) = L fi9.X 
' I 
3-63. I' - t;(X)- 2.SS(0.370) + 2 18(0.003) ~ 3 17(0.426) + 231(0.045} ~ 267{0.1.SI) + 21i(0..00.S)- 281.S3 
' V( X) - L:l<x,X.•- t•l~ 976.2~ 
t I 
3~)1). a= 67.;. h = 7bO 
a) p = f.(X) = {a ·~b)/~= f>!q..; 
V(X) =I(b-a•l) -tlft~=.;6.:t; 
b) a= 7.;.b= tOO 
~ = F.(X) = {a> bltj = 87~ 
V(X)=I(b-a .a. t) -tVt~=.;6.~ 
The f"ilngeohalncs is tlt.essme,soiltc mean shifts byt~differeooe: in the "~-o minimums{or maximums) wlt.ereas tlbevariancedoes oot cJ!.ange. 
3-71. TJ!.ern.ngeof\' isb . .;, 10 . .... 45, E{X) = {O ... 9)/~ = 4-5 
E.(\') = O{l/ tO) • .;{l/ tO) + ... + 45{1/ 10) 
= .;IO(O.lh I(O.lh ... .... 9(0.1)1 
= sE!Xl 
= .;(4S) 
= ~2..; ~ 
V{X) = 8.2._;., V(\') = 5 { 82.5) = ~062..;.<7>r• = 14,36 
3·75· A binomial distribt:tion is based on indepe:rrlent trials '\ith 1'\.,.oot:toomt'SaDda oonstant probability of Sll.OC('S.Soneaeh trial 
a) reasonable 
b) iodependmoeassu.mption not reasonable 
e) Tlt.e probabiJityt~.s t thesooond:oompoocnt failsdi:pendson the failure timeoft~.e first oompon.."'nt. The binomial distribution is not 
reasonable. 
d) not independent trials withoonstant probability 
e) probability of a oorrtttsnswcr notoonstant 
0 reasonable 
g) probabilityoffindinga defect not oonstant 
1\) l ff.l!.e fiiJsare independent with: a oonstant proOObilityof an und-erfill. then the bioomial distrib1:tion for the m:mbcr packages u:nderfi!IOO 
is reasonable. 
i) Becauscoft.J!.e hursts.eaebtrial (that ooosistsof sending a bit) is not iOOependent . 
j) not independent triaJs withoomtant probabilit)• 
$·n . {a) P{X • ;) = 0.9298 
{b) P(X > 8) = o 
{e) f'(X = 4) = o.<m~ 
{d) P(s :: X s 7) = 1-o.~ = 0.0016 
m •l P(X= 5) =[ 1~]0.0 1 5 (0.99/ = 2.40x lo- s 
b) P(X ~ 2) = [1
0
° Jo.OI0(0.99)10 +[ 1 ~ ]o.o1 1(0.99)9 + [1; Jo.o 12(0.99)8 = 0.9999 
cl P(X > 9) = [I; ]0.019(0.99)1 + [~ ~ ]0.0110(0.99)0 = 9.91 x I o- 18 
dl P(3 ~X< 5) = [ 1
3
°]o.Oi\0.99)1 +[ 1: ]0.014(0.99)6 = 1.1 38x I o-• 
0.9 
0 .& 
0.7 
~ 0.6 
0 
.Q 0.5 
2 Q. 0.4 
0.3 
0.2 
0.1 
0.0 
Binomal (10.0.01) 
D 
0 1 2 3 4 5 6 7 & 9 10 
X 
P{X = O) = 0 .()04. P(X = 1) = 0 .091, F(X = 2) = 0 .004. f'(X = 3) = o . P(X = 4) = o aOOsoforth. 
Oistribt:tion i ss.kt!'\~·ro,,;th /';.(X)= np = lO(O.Ol) = 0 .1 
a) TJ!..emost-!ikely\'alceofXiso. 
b) n.c lea.st-likelyva.ltlCofX is H). 
0 x<O 
0.4219 O~x< l 
F(x) = 0.8438 l <x<2 
0.9844 2<x<3 
3 :s; x 
where 
f(O) = 3[~]3 = ~ 
4 64 
/(1) = 3l~Jl~r = ~: 
? f(2) = 3l~H~l= :4 
/(3) = [~]3 = - ' 
4 64 
3·85. lcl X ll>."f)(lt<: 111< munb.:1 ut"liu>t:S the I~ isoccupi.oJ. 
Thro. X h"s" hinomiol dis1ribU1~n whh n- 10 and p = 0.4 
~) P(X - J) -(1; 10..11(0.6)1 -D.liS 
b) Lei Z &:nocc 1hc: n11mbcr ofl inlC 1hc: line i< t'OT oc:cupioOO. Then Z ha< n binomi.:ol di&1ril1111ion with n = I 0 
nnd p • 0.1). P(7. 2 1)- I P(7. -0) - I ~~~D.f,!\1..1 10 -().999!). 
Cj &{X) = 1(1(0.4\ = 4 
3-87. Le1 X lknote tbc n11mbcr of toomings the light is {tl'(ell• 
a) /~X = I )= [~r.2'o.8' = 0.410 
bl P<X =4l =[~0]o.2~.S .. = 0.218 
c) P(X > J ) - I P(X < 4) - I 0-630 - 0.370 
3-89. ::.) n - ~0. p - 0.6122. 
I'( X a: I ) = I -I'( X = 0) = I 
b) I'(X ~ J) - I - P(X < 3) - 0.999997 
c) J.l • l:(:\') - np - 20•0.6122 - 12.244 
V(X) = np( l - p)" 4.7-IS 
a = Jvc:<t = 2 179 
3·91. u) Dtnombl d i:lt."''blll>On. p - I 0'1.16•- 4.59394E - 06, n - I EOO 
b) P<X ~ O) = [~1'09k0(1 -pl' "~ - o 
c) p • ll(X) - np - IEW'OAS9391l-06 - 4593.9 
V(X) = np( I - p) = 4$9'3.9 
3·93. Let X dwocc tbc J!O.SS<I'S<:I'S with ticket~ thottlo not show up fll<' the lliafu. 
Thro, X i$ binontinl with n - 12S and p- 0.1. 
a1 P{X > S) = I - P (.\' < 4) 
= 1-[[ I ~S r.l0(0.9)1l! -r ('~S r. l 1{o.9)1~ .,. li~S Jo. I! (0.9)tll - [ I~SJO.I1(0.9)t!! - [I ~5~0.1' f0.9)1! 1 
= 0.996 
3·95. P(kngllrtf ·'")"< 4) - 0.516 
al Let A' denote d~e tlll<ltber o l'pooplc (out offh cl Utili w:~illels tb:ln or equsl l<l4 hours. 
P = (oV = IJ = [ Ho.s 16)1(0..184)' = o. t42 
b) Let Ndcoote tbc ntnbcrofpooplc (out of lil'l:) thot wnn more than 4 1wu~. 
P (N - 21 [ ~ ](0.4S4)1(0.S 1(1)1 - (l.J22 
c) let N d<noto ""' number o fp•'<IJ'Ic (out ofliw) thttt '"'" mu"' th:m 4 houn.. 
I '(N ;> I) = 1- /'(.V = 0) = 1- [ ~ ](0.5 16)J(0.48~)0 = 0.963 
J-<)7. /'(chon[;>'< .;,lays) = 0.3. Let ,'( - ntlll1bcr ofdte IOcltnn~ mode in less thnn 4 dnys. 
31 P<X = 7) = [ ~~~(0.3)7(0.7)) = 0.009 
b) /'(X< 21 = /'(X = 0) ... )(10 = mmtber <lf clrong~J rm:rdi!i11~SS 11Htll 4 tf/l,t$. 
_ [ 1~ Jco.J)o(0.7l'o ... [ \0 Jco.JJ'co.TJ• + ( 1: )<o.3>1(0.7J' 
= 0.028 +0. 121-0.2~3 = 0.382 
d P<X ?. ll = 1- P<X =OI = 1-[ 1;jco.3l0(0.7l 10 = 1 oms = O.?n 
d) E<Xl = •rp = 10(0..3) = 3 
3·99. 3) /'(X = 1)= (1 - 0.5)00 . .5 : 0.5 
bl /'(.\' = -t) = 11- 0..SI)O . .s = os' = 0.06!.5 
c) P(X = 81 = (I- 0 . .51' o.s = o.s' = 0.0039 
d) I'( X~ 2)= P(X = I) + P(X = 2) = (I - 0.S)00.H(I - 0.S)10.S 
- 0.) 0.~1 - 0. 7' 
el I '( X > 2) - 1- P(X < 2) - 1- 0.15 - 025 
3·101. Ut X •~:nc~~e lh<! nuntbct of IJ'i)1s II> obtain the iitst JUC\.'\.'SS. 
3) E<Xl - 111:1.2 ... S 
b) Occm~ of the 131:k (If memory proll<fly,thc C<Jl<Cicd ''~luc isstill S. 
3·1 0.>. l.e1 X denote the number of ni.Jis It> cbmin the f~1 ~sli•l lll ignmem. 
T111:n X is:. ~<<>n~tric r.uxlom vurilbl..: wltb p - O.K 
3) I '( X = 4) = ti - O.s1>o.s = 0.2'<l.s = 0.01)64 
b) P(X S 4) = I'(X = I)+ PiX = 2)+ P(X = ll- PIX = 4) 
- ( I II.H)00.l! ... {I 0.8)10.8 + ( I 0.8)10.8 + (I 0 .8)'0.8 
= o.s - 0.2(0.SJ- o.2~<0.S> - o.2'o.s = o.99S4 
c) /'(X ~ 41 = 1- /'(X S J) = 1- [/'IX = 1)- I'(X = 2) - I'(X = 3)) 
= 1- [(1- 0.8)00.8 .1. (1 - 0.8110.8 - (1 - 0.8)?0.8) 
, 
= 1- [0.8 - 0.2(0.8) + 0.2•(0.81)= 1- 0.992 = 0.008 
.l-105. let X &:note the nurnbcr uf <:.>lll r=l•tl tuabtaio • conn<'l:tion. 
il~t'fl. X is~ ~,'\)nlC:Iric 13ndom •urilblc w~h p 0.02. 
a) /'(.\" = 10) = (1 - 0.02)90.02 = 0.98'>o.02 = 0.0167 
h) P(.Y > 5> l PIX s •> l [!'(.\' l l1f'fX 2)+ I'{ X 3h Pt.\' 41 PrX ; >) 
- l- [0.02 + 0.98(0.0!1- 0.98' t0.02) - 0.98110.02) - 0.981(0.0!1- 0.98' {0.02)) I 0 00}61 0,9039 
Mnr nlso usc: the r...:t thnt I'( X > 5) is the probobilirr or no oonnoctioBS in S tri.lls. Thnr is. 
P(X > S) - [ ~ Jo.D20o.9ll' O.!XH9 
e) EtX)- 1/(1.02 - S() 
3-1 07. p 0.13 
(a) P(X - I ) - ( I -0. 13),.1"0.13 - 0.13. 
(b) P(X - 3) - ( I - 0.13)"'"0.13 - 0.098 
(~) 11 E(X1 - lip - 7.69 .:8 
.!·109. p - 0.005 . r - 8 
u) !'(X - 8) - 0.00.S1 - J .91.YIO ~ 
b) 1• - £(.\') - l - 200 dn~ 
O.O{)S 
e) Mcnn nurnbcrol'dn~ until oilS compuiC!l lilil. NQw we usc- p 3.91xl0'" 
p - Etl') - I - 2..5{i:c101" d3)'S or 7.01 xiO" ) ~GB 
3.91.<10 ., 
.l· I I I. Lc1 X denote the numhcrof tmn;nctilliU until oil oompnlrn< ""'"' foiled. 
Then. X is ucg;stn•c binomial ~n.Wm '""inble wsth p - II) • and t - 3. 
al E<XJ = h 101 
b) V(X) - [3( 1-10"0)1( 10' 1')- 3.0 X 1010 
3·113. 1\'tl;oli•t- binomia.l l':lndOOl \':iti;Jbio:: 0~: p. f) - [; - : l(l - pt ' p' . 
When t - l.lhis tcdttl'e>IO 1\x) - (1- l>l'''p. "llO.'h is lhe pd(ofs g'"'"IC:I' ~ t.111d0111 v:lriloblc. 
Also, E(X) = rfpand V(X') = Jrt I - p)Jip' reduce 10 EIX) = 1/p noo VO() = (I - p)'p', respectively. 
3- I IS. ul Pmbnbilit~· th>t color jl<'intc> will he di.ooUOlt«< - 1/10 - 0.()1 
1• = £(.1') =.!.. = _!_ = 10 cbys 
p 0.10 
b) /'(.\' = I 0) = 0.9°0. I = 0.039 
cl l:M'k of tncnlOt}' pn>pcrty implk:s lito answ1.-r '"~"als /'(.\" - 10) - 0.9•0.1 - 0.039 
d) /'(.\' :;3) - /'(.\' ,. J)~ /'(.1' - 2) + 1'(.\' 1) - 0.910.1 + 0.9'0.1+0.1 - 0.171 
3·10 
H23. N-300 
(u) K - 243, n = ~. I'(X = I) - M S7 
th> l'(:n I ) - 0.9934 
(c) K - 26 + 13 - 39. I'( X - I) - 0.297 
(d) K = 300 - IS = 2S2 
l'(X 2: I ) = 0.999$ 
3·1 :!5. Lei X &:no«: lhc uumb.:r uf bblcs in 11>.: .!ample Lim I '"'' dull. 
~) 1"(.\ ';::: I) = 1- P<.I' = 0) 
1 10 ~38] 38! P(X - 0) - 0 k 5 - Wij - 38!43! - 0.29Jl TsfJ •IS! 48!33! 
[sj s r4J! 
I'( X ;::: I) = I - I"( X = 0) = 0.70® 
b) I.e I Y dcnoec lhc numbcr of I}Q~-. necxlcd to rcpbc-c rho .U>C'mbly . 
• P(Y - 3)- 0.29:0 I'(O.i'<l69l - 0.0607 
c) On the fir>1d:.y. J"(X - 0) = [~]~~ ~ = ~ = 46 1431 = OSOO$ 
[~3] ~K ! 4K!4l! 5 5!43! 
[6]142) 42 ' 
On 1hesocood d;ly, /'{.\' = 0) = 0 5 = 5!;\i ! = .tli 4J! = 0.49&S [4~1 4~ 48!37! $ S!4J! 
On the third''")'. P(X - 0)- 0.2?31 fmm p>:1n). Thc:n:fol<'. 
P(Y ~) 0.8005(0.4%8)(1 - 0.2931) - 0..281 1. 
3-127. 3) 
h) 
3·12 
4 ! = (1.19.54 
e-4 48 
d) 1\:X = 8) = 8! = 0 .0><)8 
- X ~-l~l. P(X=O)=e =O.o.;.Tit.ereforc,i.= -ln(o.o.;) = a.9!)6. 
Cons-equ-ently. F..(X) = V(X) = 2.'%». 
-x, x 
3·1l3. i. = 1. Poissondistribntion. flx) = e A f X! 
{a) P(X ~ :H = cU64 
- X 
{b) [nordtr that P(X ~ 1) = 1-F(X = 0) = 1- e f:Xettd0.95. we D£00). = J . 
Tlt..."l'dore 3•16 = 48Cii.bic light )l'-Stsof space mll:St besmdied. 
3·13,5. {a) i.= 0 .61. P{X ~ 1) = 0 4566 
(b) i.= 0 .61•.; = 3.0.;, P(X = 0 ) = 0 .047. 
3·137· a ) E(X)= A =O.~errorsper testar-ea 
- 0?0' 
bl P(X<2) = e- 0·2 + e ·; , ·-
99.89%oftest areas 
- 0.2(0 ?)2 
+ e ·- = 0.9989 
21 
3·1;3'!). a) let Xdcootc tf.e nu.mber of fla ws in 10 square fed ofplastic_panel. 
Tttn.XisaPfii~nl'ilndom vatiah!ewithA= o .:;. 
. ~o.., 
R.X = O) = e = o .f>06.; 
b) l.etY denote tt.enumber ofcats with no flaws, 
(10} 10 0 1\Y = 101 = 1 0 o.GoG;l <o.:m.;l = o.ooG? 
c) let W dcoote the number of cars '~itbsu:rfaoe fl aws. &::a use tJt.e number of flaws has a Poissondistribt:tion, ilie()()tl!rrcnce.;of s-u.rface 
fl a,,-s iocarsare i.ndependent~ts withoomtant probability. From part (a). the probability a car cootaimsurface fJa,~-s is 1 - O.fJ06.; = 
0,39j,.;. Conseqnently, Wis bioomial ,,;thn = 10 a.ndp = 0,393..;. 
[10] 0 10 Jl:W = ()) = Q (O.J935) {0.606.;) ::0.0067 
[I OJ 1 9 /{IV= 1) = I (0:)935) {O.f>06;) = 0.0437 
~tv:; 1) = 0 .006] -0.0437 = 0 .0.;04 
3-141. a) RX~ 2)= 1-IJ\X=O).a. RX= 1))::1-
e-0250.250 + , - 0.250.25' 
0! I! 
= 0:0~6 
b) • A= o.~~= 1!!5 p;."'l' fhedays 
It\'=())= e = 0.~87 
c) IU's::!)=lt\'=0)"-1\X=l)-lt\'=::!) 
-1.25 25 -1.25 252 
= , -1 25 + e 1. +" I. = 0.868 
I! 2! 
E(X) =-- +-- +-- =-, 1[1]1[1] 3[ 1] I 
83 4 3 83 4 
:):·145. a) n = .;o. p = .;j .;o = O.!.since E.(X) =.; = op 
bl P(X s; 2) = [ ~ )0. 1° (0.9)50 + [51° Jo. l 1(0.9)49 + [ 52° Jo.12(0.9)48 = o. 1 1 2 
c) P(X > 49) = [ !~ ]0. 14\0.9)1 +[~~ ]0.150(0.9)0 = 4.5 1 X 10-48 
3·147. (al (O.S)1l -0.0002H 
Cbl c~lco.s>•cos)• = 0.2256 
(c) c;2(0.S))(tMl1 ~ c;_:(o,<)~CO.~)~ = I)A 1&> 
3·1-19. l<:t X t!.mutc lhc numbt:t• ol'n"IOmir~s nL..,.I<il toobt:un. s rec:n ligln. 
1 hro X i> a jlcOOu.'lric r.tlldorn •oroble with p - 0.20. 
a) PCX = 4)=(1 0.2)"0.2 =Q, IOZ4 
b) Oy in~epcntkn.-c. (O.S) "'-0.1074. (,\I so. P(X ,. 10) - 0.1074) 
,;.151. Let X llroote the nwnber of rills ncc&.-d to detect thn:e tmcltl\\·c~ht iX!clc~~ScS. 
Then, X i• • rtqyrth'C birl(lmial mndom \'orinhlc with p -0.001 nnd r • 3. 
3) E(X)· .1,-0.00 I • 3000 
b) V{X) = (3(0.999)11).001' 1 = 29'>1000. TherefOre, a,= 1131.18 
\ . I S3. (a) ). - 6"U.5 • 3. 
P(X - 0}- 0.0~9~ 
Cb) I'(X <! Jl - 0..5768 
(c) P(X :;x) ~ 0.9.x = 5 
{d) a' - I. - 6. Nut appruojui:lle. 
3-1 SS. Let X llroote lht: number of c.'lls thot ~rc nnswcrcd in 30 sct'Oflds or lc!S. 
Then, Xi, • binomiol r~<>m ' •.ui:>hlc w~h p - 0.75. 
~) PCX - 9)-[ ';](0.75)0(0.25)1 - 0.1877 
b) r(X ~ 1 6)- r(X - 16) t P(X - 17) • P(X - ll!) • I'(X - 19) 1 P(X - 20) 
= [ ~ J(0.7s)'•co.2s>' - [ ~~ )<o. m'7 (0.25)1 + ( ~~~(0.7s>" (0.2s12 
+(20]c0.7S)t9 (0.25) 1 ... (20](0.75)!0(1).25)0 = 0.4148 
19 2!) 
c) J;(X) - 20(0./S) 15 
3·157. Lct w d..>noW 1he nwnbcr of calls noedcd to ol>toit1 two answers in le;s tllsn 3() seconds. 
Then. \II h11s • ncp:h-e bill<"ni:tl di<trib<lli(Jn wi1h p 0.75. 
~) P(W - 6) - ~~ 1(0.25)'(0.75)1-0.0 110 
h) E(W)- rip - 2.'0.75 - 813 
3·1 .S9. X is o l'k~oti1e binomill wilh r - 4 aud p •MOO I 
6<1() r l t> 4 /Q.aOO I 4001)0 n.'quelti 
3·161. l..ct X llroote the number of iOOiviJ:luals that rero1-cr in coe 11eek. Assume tbe itX!ividuals 31"< ind::pcndem. 
Tl"'tl· X is 11 binomial random •ari:oblc with n - 20 Md p- 0.1. 
P(X <: 4)- I - I'( X:; 1)- I -0.~70-1). 1 3 '1). 
3·16.'. Let X ~:note !hi: nurnbi:r llf:dS<:mblk:i ne<xlcd hH btain S dcfcctivi.'S. 
Thm, X i~ ,, no:gMi\'C birl'.'minl mnoom W(f"inblc ,vjth p - (),0 I ru>d r ~ S. 
a) E(X) - rip - 500. 
b) Vi X) • (S • 0.99)•"0.0 1'- -19500 ao:J a, -222.49 
3·16.'i. Rcquiretll) · f\2)H(J) + ti4J = I. Tha:reforc.c( l +2 + 3 + 4) = 1. Thm:f~,c=O. I. 
l -1 67. fvlO) (O. IXO. i) - (0.3)(0.3) - 0.16 
fv!ll - (0.1;(0. 7) + (0..1)(0.3) - 0.19 
1.\ (2) = (0.2):0. 7) + (0.2)'0.3) = 0.20 
}~(3) = <0AX0.7) + (0. 1)(03 ) = O.JI 
l ,v<4l = (0.2X0.7)-r (0)(0.31 = 0.14 
3·1~. 
X 2 5.1 t>-5 s.s 
f(x) 0.2 0.3 ().l 0.~ 
3·171. let X ~note the number of <ITO~ in a ,.,.;terr. lhcn. X is • Poisson r:ul4lom \":uiablc with >. - 0 .32768. 
al P(X > 1) - 1- l'(XSII - 1- """",..-e_,..,,...I0.32768) - 0.()433 
b) Let Y d<oor.c the number of =tors tullil on cm~r is fo111td. 
Then. Y i»!'.~mctr"' mndom v:uuhle and P - P(X l! I) • I - P(X - 0) - I - c4 .,..,._ 0.!1',14 
EtY) - lip- 3.58 
3·113. o) llypc~ctric rnndom l'nrioblc with N = 500. n = S, ond K = 125 
. 
[ I ~s~.;;s] 
(0) 6.0164/110 - 0 2''7 
J,y - ['~) . --~ J 2.SS24El l 
lm][ns) I .! _ 1 25t8. 1~5SES) - 0-'?71 m !.SS2SI!I I 
115 [37S 
2 ' _ nSOt$71 88751 _ 0.2647 !,.(2) - l s~ 2.5s24cl l 
[
12S 1375 
l (JJ _ J 2 _ J I7750<10I25l - o.osn 
.v e~ 
1 
2.SS2~ EI I 
[12SII3'51 f (•I)= 4 I = %913iS(;liS) : O.£J1 4Z4 
.\ rs~l 2.5524£1 1 
3·16 
L 
[12S][37Sl I ISl = s o = 2.3.453£8 = o.ooo<n 
.v f~J 2 .. m.w1 
b) 
() 2 J s 6 7 10 
11~) 0.0546 0.1866 0.2837 0.2S28 0.1 463 0.£15N 0.0155 0.0028 0.0003 0.0000 0.£1000 
3-1 75. Lc11 dcno<c nn in.tcrval ofcimc in hours God Icc X denote the number of mes~ thai nrrh'C in tinlC t 
Then. X i~ n Poi !SOn r.>noom v.vioblc w~h ~. - I Ot. 
Tl~en. P(X - 0) - 0.9 nnd c 11'-0.9. n:>uhll'l!; in 1- 0.010.$ hour.l - 37.8 ><eondi 
~-1 77. The binotni:!.l disu ibutiM 
P(X - x)- { n! p'(l- p)"" 
r! n- r)! 
Tltc pt~ility oflttt <''1.'111 c-.ltlllc ~fl'I\'SScd 3.1 p • /Jn 30d chc prolldbilicy mass function cnn be \\Tiucn 35: 
t 
I'(X = x) = ( " ) li.lnJ'II - Ihlnlr· 
r!JJ - r! 
P(X _ x) n ><.(n - l)x(n - 1)x(n - 3) ...... x(n -~ • I) ,L (I - (i./ll)y-
11' x! 
Now we can n:-<:.~pl'I.'SS a5: 
(1 - (Un)I~'- (1 - (:Vn)rii - (:Vn)l" 
In the limic ns n -+"" 
n x(n l}x(n 2}x(n J}. ..... x(n x • 1) "I. 
n' 
As n -+ • me limit of( I - (:Vn)t': I 
;\I~. we know tho1t o.s n .-. 
Thu~, P(X X} ' A ~ - .: . 
x! 
Tlte Jistnbulion ofllle pruiXLbility a5SO<'Ioted with this IM'IXC!S Is kn<l\\n a> II>? l'oill'(Jn disttibuti.m :md we can 
.::<pn:ss chc prob<lbilil)l mnss 1\lnccion ns. 
e ">.' nx) - :......:.:._ 
x! 
3·17 
3-li'J. E(X) = I(« + tu - 1)-.· •.• - b)b- u + II 
tt-E~ l l>(h + l) _ (n - l)ax. 
_ o•t I I _ 2 2 
a n 4 1) - 0 a • IJ 
(b' - u· + h- al 
> ' 1. 
2 =l(b+ n;(l> - a .. IJI ~b-a- 1) 
(b a + l) 2 V' 
(b - a) 
2 
b I l:!;- •;·1 
V( X) = -"'. "----
• l:tl 
I • 
• • 
"' (II - a - lXII - a)· (b I n)1_j I 
' , 4 
h - a - l 1> - u - l 
= b(b + 1)(2b + I) _ (a - l)o(2<t - I) - (f> - a)l b(b - 1)- (a -1)ol- (b - o - l}(b - a)2 
6 6 2 4 
. 
b - a+ l 
' = (/> - o - 1)' - I 
12 
3-181. Let X - Mrtlbo' of~SCOS''fS with tJ n:s..-n'-"<1 seat "'ho arriw til• !he Oi~•t. 
11 = num~ of sc-..u m<:f''nti<tM. p = probability thO! a tickctro ~~cr oni"~ fort he fl ight 
•l In lhi• pgrt wedctcnninc n >u:h th:u P(X ~ 120} <: 0.9. lly tc~inl\ lorn in Minibb the minimum v:tluc i< 
II - 131. 
b) In Ibis jlQlt "cdet,·rminc nso<h that I' (X> 120) S 0.10 whi~h i><-quha~tlll(> 
l - l'(X!: 1201 sO.IO Qr0.91)s P(X!: 120). 
ll)' tcstin~ f<1r 11 in Millitob the :K}hll ion is n - 12J. 
c) One po;:iibk '''"""'' ~llows. lfthc airlmc il ml>it ~'<lnc'\.-mtxl with losin;g cu;uorn<r.l Juc toowr-bool:i'lf!. 
thcysb:lu1dotii)' Scll1 23 tO:-kctS for dtis Oight. The probOOility o>fo,~r-bookins is lhcn31 rn051 10%. If the 
~itlinc i~ m<J<~t ~~>."eJl'led " 'ith "'"'in !I~ li•ll flight, tl>ey soorrld ~11 I J I tickots for this flisht. The chnl).'(: 
the tltght is full i~ tl:cn :11 h::1)1.90~\. ~t c:dC1Jl:.tticans ;,U,.umc custonrrs: ::~n i\.•c intkpc:nl.kntly on:.t group~ 
ot' 1"-"'Ple INI ani\ e (Of d~ no>l :liTh e) tO{;ct1.cr tbr lr3\'l!l make the On:ll}'Sis more cocnplbtl'<l. 
3-18'. lfll.,lr>t >i7e i< sm.>l~ tm•ufthc lnl mi!!fll he in>Unicic.'nt rodetoxt oonconfonn ing pn>duo::t. For exumplc. if the 
lot ,.ir.c ES 10. lhcn n );,Ullpli: cat' >-izc one has~~ probability or cmly<t2 of dclccling ~ nnnoont(nmins prot,luct in ~ 
lot lh:11 is 2~ u.c.ln .. -onfb:rning.. 
If the 11)1 si~eC i:s wgc. 10% of the lol migll4 be a IOfl.!CI' 511<nple ~zc th<ln is pmctic-~1 or nc«:!Snry For tlUIIllrk. 
if the 1<11 $Ux: i> 5000, then u $nmplc: of SOO i:s rcqlliml. FI.W1hcnnOCX', lh: l>in<>minl "'J[X'I>•imation to tile 
lryperscoorcuk disuibutiun,..•n b.: used lo >lr•w the IOll"wmg. lfS% oftl"' loe o!'sw.> SOOt> is rwnCl'011IOtnung.then th: probability ofuw oonconfi)Rl1 in~ [X'I)du..'ls in th: Mmple i:s (lf)proximntely 7 x 10 r: . U$int; a somple of 
1<10. the """"' pmbobility is ~ill only 0.0059. The >.>mplc of s ir.c 5()() might be mucl> IUIJ~c:r lh>n is ncxxJciJ, 
3·185. Ltl X &.-oote the nurnbcr l)f rolls 1H\XIU<'I:d. 
lti:\'C'lruc :tl c"ch demand 
!.! .1.00!! 2@1 'l(lf.O 
0 < ~ ::; 1000 O.OSx 0.3x O.h 0.3x 
rn..'31l profit - 0.0Sx(0.3) + 0.3x(O. 7)-O.h 
IOOO~ x ~ ~00 ().OS. O.J( I(XI(I} "- O.OS(x - l(l(lO) I 0.~ .. 0.3x 
me<~~~ J)fofit - 0.05•(0. ,) + ((1.3( 1000) + O.flS(x - 1000) 1(0.2) + 03x(0.5)- O.lx 
2000 :S: X $ ;1000 0.05x 0.3(1000) .. 0.05(~ - I 000) .I 0.3(2000) + O.OS(x - 2000) 0.3x 
mean profit - 0.0Sx(0.3} • [0..1( 1000)+0.0S(x- I 000)~0.2) + [0.3(2000) 
+ Q,QS(x - 2000))(0.3}+ 1)3x(0.l) - O.lx 
3000 sx O.OSx 0.3(1000) .. 0J(2000) + 0..1(3000) ~ 
(I.OS(x - I (l(lO) O.OS(x - 2000) 0.05(:< - 3000) 
"''""' prolit - O!ISx(O.J) + (0.3( IOIXI) • 0 .05(x - IOOO)K0.2) + (0.3(2000) I· O.OS(x - 2000)}(1.3 
+ [0.31.1000) +O.OS(x- 3000))().2- O.b 
Pmfd M:tx. pn>lir 
0 :;:: ~ :;:: I 000 0.12S x S 125 nt" - 1000 
JOOO < • s ~oo O.o?Sx + SO S200 ot x 2000 
2000 $ X $ 3000 200 .Sl()l) fit X = J000 
3000 $ X 
-O.Oh + 350 S...'l()O m x - 3000 
Tire b>k<rye:m malt.: ;:mywh<n: frum 2000 II> 3000 :md e:rm the !l:lmc pruliL 
3·19 
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