Finol Exom
1'{A}The ptab- ihal an'Americon industry will lccote in ttaypllli: o.7 , the prcb. tlrot it vtil! lctole iti Brusseis is o.i ,' ond prob- thot it wiltlocote)p-eitlqr Munich c,, arffiiil-l o.8.whoi is rhe prob. thct t!.re incustry will locote inL'both cities ?. z-Mtyn$2 3.neiiher cities ? l.Munich siy=::!:!it has npt locate,c in Brussets ?t'l/E' ;(-r-,)t)f A,B,ondic areindepencentevents,show thoi:"Thetwoeveiits A'and BUC ieiisoindependent ,,.-.
\:=:-':'/:
z-{Afind the meoi p and lhe standortl deviation q oy d" nurnber oj ie!-ec.tiu.es X in o somple of 6ao pans' mr:ce by o michine , when the prolt- is o.o4 tr.Jt afiy one o; tnu prffirHtffi" .Then find on estinate foi .each ane ol tiie follawing tvvo probobitities: (7ibilx- y-l-_g end (z) pi 12 . y .:38 ) .
(B)The nuniber ol spotted imperfections in on, ,#rr", ,,::5,1, , poissqn nanndom voricbre wlth cn ,r#r*
_ 5 lrnperiections . whot is the prob. ,, y:i: ", ry
,ff,{r.r"a;onl,h 4 mir,ntes 7
--'-- /\-- L
\thot any one of them );lt require repcirs , what is t!-,e r,tcb.
repoirs ? , (Z)oitt olWobricoted rob.ots, er;octiy Z to require(3)ti*74-*bot is the 5h one to require repoirs ?
v-=-4 'r<:jig)Cansider the hypergeometric c;stribution vtirh : l,l=12 , o=1 ond n=3 ; vtlti:h nleens that tlie toto! numbzr t;f
u;:iis i; L2 , the n"fr5€ii;S aefeaive units is <Aia the scmple size is 3 - vvhot is the prob. af gettinE :
3-{AiA ccmLlony fXbricotes rcbLlts. tf O.1 is the prab,ttlat:(1) th; /" robot is the Jirst ane io reguirerepairs i X=j$ .^-\
(l)n.o C_e!:7t,ve units ? (2|ot teost 2 dzfective units ?. Find the rtean ei.the distribsticn by tuto methoCs-?
o
=s-*ej. Jirt|.ilsprc.u: !9s1i-r1.ngi?,r.-.. . . .
cnd (3) the mean of l,=2 + j * by *to2 D( E ';6*--zI\"y'X , Y be tu.tc independent ranclom voriobles
,Find:(lJthe moment generq.ting funciion ol Z =(z)the votue "J@D (
sucha
3X+,
tho2Y oi z,
. s/A)Let X ond y
". r:f+. {z)P(o. x :hove the joint p-d-f. f(x,y)= k ,Llv=!1 , 'p)E(x fz$,fGly) a:l.ct IUlx),
x<L.,O<Y<1).-zt
0<xcy<2.til , ond
Finci:(i) f LG) ,
@ P(o <
..::/#-r,;:. - _'rl'_ -'
Ui*l.- Y{
Let tty(O): ezo' otrd.
inC:(7J Mrr,(0),where{31P(w < -B)
cr)
l{:-'.}-if.a-.:-: --.---.r..-:i,::r; ir;.;:rj?i:.It*2;S.:.;.;qts:r?-$-?-t*_7Jg.T.45|.A?.vf-ge.:W.;5t;r:;:,,fl-q,.-";i..r.._--..._--:.._?.-:r:--;1</.?-,1,Tr.r4-;
' l-q.-\2-"-\2
y''lvrt,'cl=--*f ) Bflts1./s=,.'.? 2- -\.
e;fL"n) Mr-ic h- of ':Brq 5Je/s t-s
;
6t"n7
e
' *c*i,
o6
I
P (M - 6l = -K/41
o'.1 'P(A/ - g)
=.o-Y
- r(N //E)
'o.-3 = o'\
@ N"rllt- ^'L'I- -.-
-" 'i
PbtiB) =
0 'g iu.*+-./-
-Dl
p(tlo)-=
o-V. / t;l -
t=
O 64Y C,'/;us @ Nu r/ r'c /, "X!/- €) q^e,.fl,.. c,./q
@ rluniCL ji/e** ryf ,'/ A'.s 4-.^l-r:Z-ssets ?tu\-/- +^-,- 4
P(^'t)-.-T ? ('Li -- " 'Lt
P( P\ oB).?) So\r^ c. \
' ,lLt_qB) = PttT) r/( 6)crE : o-a t o-y P(H/18)
P(r4nS) t o-1 +o-t-{ ---B. o'l
* rWfrsl = t- ((rtUB)
<>-8 ='o'Z
Phrl -?(HnB)'9
/-P$)a.
o-n- o-J
l'L -'t)
t
aJ \.--,,'o\
tt
S o.*t\e +
gr/.u Po"d
//u*,y +
? d) @:r(rzq^ <s6)
rlt ?
P.
F,^J yL- i*.,-* a f---J tu- s\.^'','d*il J e''/io{'o^-
P:l .'^qq. 1*''-"'/es lo--,-f, / ".uLa^-L
o P (|r-/l,, t*-Lt"--t
:.
) d'
(oo 2
, A-7
o-o1 -: 7'o 7d
-Lt
(oo Porfs 5/L
n), f,1c. h',-z
Pfo- b
,,n 7 r*4DU--
d' Bir**-&
B'**J F0 x) { 7'-'ttP =-o17. r "-lf,
Lcr.u e'v'-b"^*d
36t( llNlz'-/*(:1*-
N
-l \-7
6-oo*o-ogT I o P(tz-<r ------------, -.--I (^ , ,,
>3.,) <rl'7' I
t'\ f,-r c'- = 3o
,, li', y = j:_/ -i'i'{-'a?t-
I @;-1o2,
? L/: Ko') t
G) f U!- L'-1*-a
--'---
LJ
= ./ 't Ka'36 -,/1
>Y.-y'1 *,*
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V =< l-.s )
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nve,at ) trnr--'
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t2- TerLca
+
Y-
d\^J is'
t.^ 1-t q". irr.^rF-e- S
N"s t /-o'\------.,-------
t^Per
)
_s-Ld:5 T-_) 'lo,w-,i*- P ( K 52) =?
:.r.Ln- \ r \\^4\r.^--
t)
C!- etz n1t 2)-
ffiffir;::."..iq>:*:4,#7I.ax.:.--)^1;.1-.-..-!::.:Ij:ii1a|:.',!:;::l,.,!7.tw,aw@@ifuP.8*
?r"\" -+ s p"tt-; j--\--\tu
(^ -r I n=o, l,tr--
5+o-\ -E
: "'j-':11 "'
a.t3g3 + o-27"( r6-Z7o(o'6n6s
f (r.) t ?r,)
(r)' ."-,
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GiS5,t2(") r
(2)" -2;l- et
ol- =l
4t .t2l -t
-f=
t) '=
-:
o<-T.
\FJ;-\qJ ?=
Qr"V
oJ
o-|a
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J f t
0
{ c4 tlivv
1 u t- i;l-v'.rrts
.
.. /(>
?=-\_\,^t.-...--
. s-l
K=L
G.-*. Vi c
- Iv.1' fet"\-s is
1 t ?. t rra c"J yc, l,.v s ,
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r :t <{h-t,':-'./-d---f^. 4eor^e4f,-C
',
4;--- le G; ^"*;,,.(
o@
@ t'k
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i
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o'l ) 7 =o'1 -, fr=1 , VZt
rt -4
x
5x
ff), f(:) -F.'.
6. lz tt
X'=-ti' ['}(x =a) =
r-l7 .PI: (" 1) ( ''l)
e X..c{/.rJ
.+-. Y "7,
+\)1 ':'
=/IoJ aFF
2 l- o rq4 v+i -€ruti,/- {<p^irS 2I
:
-,: V t"r!-L.t- J\
6251, d:l
\L- ?t-,<:E
lt
(ena;ts 7'r,ft \ ^-- +
l
|lsrafffi i: :.:,_ : - -:.+:--s.Fj?--'
@--f '^= X
..'.)r { -sl/rx+)=( , )F'.lu'----''- -t-+
' ({; ) i ?', o-ooc \ Zt5
5* -b46;'o-'^&
rc - Zal?-----a
@,uL:---
ry:! , 1--e-'l"t-? ,o- 4 .^* t
O.{-.o|ue- = t-l
S-bvc *ry:- = 3
S>L:.:
i?rr)--'_--
M: lz ,,Q =q,6_,5F = Yro d.fl.di*o- = o
P6="1(; ) r I )
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o
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(u (!F)
)
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a
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3
ttrl/f l * o- ol
lz9
-Z
(
:)GiI\
(I
/. -5 r ttx;ail
:- dfal r lf(t), z?(z)+ sP(sr'
G+
{vi tz t
. f(x-l)= \tJl'i -o-so'
Itz \
; o r o.Srl r 2("-2/f/)r S ( o.u/f/f)
rdu/
l? tz.
\u 1\< Yrob '1 t^a^,c"rd-Cn
\-r^v.-! Y-V N
i
*o{n)= 5jn, ,*,,
rS
fu* P*. \
o@
6P (z<
Neo ^
f,/t*t *+ f
.t'4/=&-
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;
l,rfrA(o) fo,i_1.
^tjGrp-^z*.rrc J.r2O
"tr'---iz
,a*
l/s',tS
r,@) =
5 -qe
' O7t
/- qoJ
/:frfria
\
Het'a" J
- ,- a'-', ,? ;. =F^ r tl * $'rtl - (*^r t 11' = --.'
-w€?,i 9 preli-^C C*_.lr*( 6,.,) + __, v-i,v =___,=-U,I* -, e
\=--- f ) 6t-'<)\/---,O'?-,r.n.t\ f,'q
\ -t 1 .-- J\ -L.-. \.t^tv +)
,E?ellre%,
--f; (c: )
5 -'i o +{ b
o lz-ry'<-
i
b
(s- q o)' (5-qa)'
+3zol) Zoc, _ lb
4rc'- ---
--?-\-)
+'(t)
( rcj)' e,
[t,i (t-,ta) j -Lq@])/ -.r4\ 5-qol'
:s -_w1(r q)' t
'
:I
:
-LrlL,"-* (oi=
-E(s-vo) I (s-uo) , ( s -vo)
2oo _ 16o
( t)'
--."-t-- '-'!- r',\qy-r'o*.r.Cj_ -:-\(x ) = cY :
:'.' "'..-., -i-.-i..:;
- - -i-rr,1:*.::_:_
.:,.:. -,--...: -,. .*....,...
..
oo, P,^ / gtt*/r;c z C { -2 5, ZoIIv
ou4 r
IN
? (z <x <s)
=@Lt
{
? ( z<
Pt"r- s)
:
Pcr :
P (r<
(l) ' (+))--
x<5) "
P (9)oP(,t)
: 2'-' P
= (i) - i+l
L{G^'^"\r,q--1.
\_l _.4r.{ P
ae) : ?(s)rf(q) =t{ )- (\
l- a '
-
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o -\L${ o-\o2.1 (
L:--z-
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aa
a
t
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f"n HuTA*]5
[(/l , 17
\z --zti
f(2rsX) =? tf,!) = 2+t{'
oo. [( Y), 2+ 3 (s)
0J" k'^o-
^-/ -4/ -J'-"/
I
2 ^d N"fbJ
-
Ll 5e 8,^o^,& t'/elL*d
1 = t-\/,rf = 'l 5
h
/=,tt P6, y'
-'p -)t
@fuF) = eb'P^rol
a
'L.t 3*
=-*6
j'Ot=
sqL Gl--*@zs*b)= o' -( ! r*o=) , ?/Aeyv y6
'@1--..-:.-::..--.--_- -.-:_- --- - -
as)
C . H
a,
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o
@
[,3 Pa P F(r,J)--x o(x(J(z
)
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2
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foi^\ , l,J
aJt/2"
.,lfo^llJ9 ,=,
1.
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. L Ja
*r =l-;
[a-\-] f,?' O F (','11 )z os^t @ Jlft/,q) J^Jy=/.zt.o 12)J K J^J1 ={O U:
a
,
f rr, -trI It,t t - )
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'(r //,) . ,f ( r, Y)
f.cY)
'-/ /y) ) /( r, y)
ftrt
ftr, j) ,JJ
\ ,l'-- L'J I
J1
?(q<^< t ly=-;"?(x//)=
1,5^/yl ) =
ly,r-a-
z- f . rz 1t/t-7t- ?- Jo
-t .'n;2L*l_rz-:1"El-/
, t.),/J
t 7_/ 5/_/ (-
L
: [r J*) L l
@L(r-\)-----
r--\-z---:
) f, ([l=t3:Jl, (/)? ( r lx./ z
.o
:. !nJ")
f (q< ^
<,.lr,Vu)
ir., ='.{.') .-Jx
.J()
I
--?xllslJd
t/z-------1
9tIZ
: '/'- = lH-
(-,-4) ,{Gr)
o'. 3
Jr
o { X4z
tH '-'@ E (.xf s/")
€(x)- /nftrs.)XE (*1,r.)= if frxl V.
o
71,r=/ x.+ Jx
_ rzt\qi
I=.-i '\i-.-= o({(z_i k-xt
-1 : )\_.___
rdt
-\,-/,/
*
3r. ,'
L
/ -,-
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r9t,
t1
U
I
5t
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L
B
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o
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=[;, +. 1,,
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v
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r)
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