Modul4
Kandn Vae and phoese
Litaoduchon
tandom
be
Aaug met
prdt t atual
vale the Vale
mot
the
frdit the fohese valee
thon
atatia te atdis, hhee paafas
meas distibuton
Aeh as avug pouer
fooee Can be obtaned.
¢
evet o
Phas abaly
dhis
Qwndng the wmbe of
3t
ootaumd by the total mbes
©tcome
oUt cemesi
atpaoaehs to the
2
mtms thn
Rvent A'
a
s hlatd to mathenatial axo
Jhis Aegauded
(statemts.a tacpottean'hich ty
at alslitl , acefptd
the) wten eutiow
a dsndon
Jhus n qeueel,vasco ttiat
t are
Cipinent OCus, a h l be
butcome called aa aanpe
the kth Benbte this the
pomt, heh
dtuntoy vede.
utmed tho
) Jhe et d hieh
Lepanient
taeofpondu to ettur a
R a seet aledan lemtay
aaple point
(3) At ane
Rvent. e euent
afae s' g caled
Jhe enie Aempe
(k)
wet. exelie the
wo sent
(5)
event
HI- 2
dlis ctte
Cewntale
kte utcomes
Co.
dhe contnus
Sample apace y a
nae
meaguedsed ot the ontant funto ttat
maswe uent A
Afnebaboilisy ahve mbe fo
-
tat
(ax):
P[s] I aclue theu
Aand B A B
3) 9f he
ale
P AUB] euet
below:
The leto n M
c
to the
Jhe the aows and taei sletioahip
aham n Fagc2)
benn duagram
Veun Dagram
duabra nheih
etatn
shews the
9et
botuseen
p t to 1
wth total. a
deid
aon
-(4).
I- P[A] teo eUet A
A
aa the Coamplnet utally
B ae net
A and eveut A/
wenta A
(2) ohen
fhe
wlie na]-)
[B)- PA
P AU6
P[AJ t
Jont
Elmentag
Phob
enent A& B
MI-3
ae
3). 9f A, b, melude
cesie eet that the hendon eepelnet
all possbe outcoms
then:
PAJ + P[A ] t
Condutnad Probabaditt
demete oceed. IN
dat P[B/A]
P
condtonel pebabl
the ewent
event B puen lalled
A'
B'
P[An6] -
eent B P[A] eient A 8
ocned
ohan A hay
hohe
Pfane)
eforsd
can be
tapallky
Con ditionsl frsbahley
Jhug Jonnt
probably
denenty rel
Lrnt and
atso capnentd
(0
helahn
Baye's Rue
hat the Conoldonal
ta be
to clenetay fnotatbey
event
()
P[e1
cndes tis Codtion
-(2)
Ao that
euent A
Con
to the elntat
heum
fund P[Ao bo
Bo
A
-P
Soton
fheri robabults
(0 Jhe
eseut tsonaallg
olenot
oheie Not that
0 and 1
ere
(3) dhe condtned frebab iy
dnt the eiewto
alese o and B,
and 1, Nspet
O þrobakudy
Pstenai paobablly
sent;
| Sent
’|
P[al6,] B, Ule
Bne the eienh
|-P
2
(-P]Po t PP
P[e, ) + (EPP
's Rule
P[BolBo) (FP)
+ PP
HI -5
PS,) e )
A
P[B,3
(-) PI
Po +(-P))
Radu vauable dusuhe t
wsed t tcome
Rendo vaiebles to the
mbe
randon cxpset. valuste te atomes
f Eg: Agring a sage a com
random capimet ray
a tail
tae eapeemenl X
Jhe outcame
coe
anon expinet by t
p t ulan
uler out hamdoy
The fat
one of we
be
wtt Aane
tepunent
vareabe
Jhe
bttn (15)
melehi distuletan
called
whele funchon fel)
distubutin fnchon
funchon (c'ds) d
Jhe cmulatue distubton
funchon Pxo) s bandas beteen
) Jhe dis tubtin
monolone
dste buten fnchm Fela)
2) Je
-16)
((6)
2| <a
e
3) &an
2
C
fx (-0)
distiebutim fncben
Jhe dewatwe
d Exx) (1a)
bx (x )
Galled
(PDE )
fa(2)
(Is)
-((4)
fx (2)
(20)
denis ohe
E
cal atatis
dz )
(). -
erpetd Jhe
vale Mean:
veuebk Jendon mean
Avages hal Stats
b
b-a :6)Fx
7<a
cmlatie The
teae fntn
X dithubatan
tx6)
pretoa, a (a
b)ntnal kte tubutd ds
molny beto said Vouable
X random A
tebutan dis n g
-6 M
Fnchoin d a handon vaiabe:
handom Vaeble
det 9(<) be CA
But y g )
-(2)
Cpted value
(2) Ao
andom veiasle X
fanction gx) a
edal Random vaabe
the
An the taval , )
Cos (x)
et y 9(x)
2
ottese
9) .dx
dz
Ces(2)
t(y) 2TT
Sina
Qawnstion
squoe Jhe
vahanee
Vaenee
mproeuid Jhis
atandard Called
oas
Dewnahon Standerd
varabe
x andom Vasiane
2 X
moment cenal d
colled
[(<-H.)"J
: E
f(-)"xl)
() - d cetiel fh n
Jhe
by emet
c4) dz
m=2
he
vaiabe Tandom
-(3)
vauceble
X Tandom
)
Can
mmet an obt
opeual the
etutn
Momeat
-7 M|
Che byshw ngaly
J4 Atates Hat e a þosetue
pesete nnbes
2
that mean
and
frm tis ngnal Vnsialse qe a parta
andon
Vaame
a dit buon
dsaghon'
RANDOh PROCESS Ho atstal analyas
concen Hedasaihan
A bac nd c h o , tilevat
o Comm ahan
andom degnalo ch as nose .
data and
2 profetes
Ksudan agmalo have
hncto et tne
chiinatn itval aeae thet before
the desabe
pessibe to
Jhe
ase
expumnt, in notoôceed
(2)
oun e
that hll fchen
s a
Aamnple apae
nchens time
he
atocashe proces
random of
Caled randon epnt op
Cone
Expemehon Sand
tcoes
the the
os ttese evento
the fnobal'aletes
by the
MI-8.
accedane wt
the le
X(ts) obsewahon mteral
whe 2T the totol
He
ti't'
Callid
ves
funchon X(4g) random
ttie saule funcmon:
e dente
j ) = X(4 })
a Aandom brocs
dene kaudon þhocers vereAte
wacd to
wsed
andl hadom
X() andon bros
Jhe ditene blw Tandan
the outo me
vaniable, membe.
random
mafped to wteone a ando
espuinent tthe
Tando wto Q
Jo a
nt
time
Couelaton andCovnane fnchons
Mean
Cenades Bro es X)
a andon pros
þrocen x(4) obtond bt
Jhe bandlom vaiable
the epectaton shrny
dbsewmg
Mxt)
tue time t
the fnoea at
Conatat
anlon froes
) x Contat
No vohanee
funethon procen X) rando
Autocalotion foroduet of 2
ypeetotion of the
sX&) Xt)oatond bt disewmg
xt
Vasialdus x43 nd
at XCt,).)
e
Rx(t t)
Nantom
vaabe X(t) and Xt).
Autocovaame un chen
Jhis
denstd by CxCb,b)
C<,, ke)
Rx(tg-t) - xstaddenany nando
Jho 4to Cevalanee
te tne daffene
poes X(H) depodi ony atoColah
and
ean e t a nthe
ta-t). we kio
that
frocea autotouaiane
atoCovoncane tanchon eztina
coletn ond dependme
uto omalye the
B to
Not Aod
tine aeceo data.
rtaete esthe Covaanne
cova
Auto covasane
fmchen timeBointo
atfau of Corela
wtth elt
chon
Ado
Jetaeen
aton
ale t
hapatus [x43+)x}
Rx lz) E
te caM be
|. Jhe
7=0
obtaunud from Rale ampy by puttng Ze0
Rxto) E[x*)]
celatan Antion Ru) is
2. Auto
Rx(z) Rx(-z)
bncom Rxle) has to
3. Jhe autotaneleton
at
maxwmm
|Rx(z) < Rxto)
Ruto coelatnfnton,Rx()
Phagoial agtaane duscubung
tee ntendependne
obtaed by obsewawg
nandon visabla apart.
tisp
to tnes t
eemds
þroces Xt) at
Aandon
o sCoelaton Fumchon.
proenes Xt) ane yt)
Cennd tio Aemdon fmdns Rxy) and Rylb9.
the autocletan
ts
wte coelatin fnetn
Jhen Chos
Gauman proes
Cownde nandon roen t) te
teal b/lo [o,T]4und eppn
ppox thn
Aanlom froes x(t) by
g) x()-dt.
Walue
valebey
hendon
"oad the
aei,
andon vaneli
dustubutd Gauman pros
Gaumam a
heu Xt)
hes a
the faun
RendmVaies tnhon ho
Y,
þneboblty dunstiy
- (-My)
handoy vaeabley
whee y Vaane vauable y'
rendn
Gawscan 2e0
and
when
Kese
A hare a
motwelied to o shoun by
A
varame up(-)
tyC%) V24
O-2
N&nali2ed Gaus an
2 3
-3 -2
dugtubtan
Gasien Proen a stab
Phoperles of a p ud to deelpd
procs ylt)
Poo 9f a nandom
Hhen the
dneat flta also &aursan:
anples
at the varables oot
andon
aet by obcewng
P(2) lomdu
x(tn) obtoued
X(t ) , x )
Set
a
radom procs G
arsan n
Nandom Vawatyle
densety faachons beng
Jhe n a set
xc)
auto ovaae
Caltk,ti) :
K,à = l, 2
e statiay
y net Ae
tten haas Xn) obfemed by
varualoleo X(), X(t,)-- wnconeletd
P4): 94 random t , t . t y ae
X(*) at tms
then efct)- xD(ti - xt))=
Jhn the random voaabsaeatatinkaalet
suasles ae mebpudet
