19091104661
Sa
H vv
an y
U
72 ni
17
STAT 330 Mathematical Statistics
Lecture 6
Oct 17 24NAT 10.21 AD 11 00am TEETH
52 I
3.7
Probability Review 2 1
heel
TUTDYER
Prof EAMEDEE
FEVEHEIFFE
1
FEEDER
A ITF a
Midtermreview ahh
1 1 Probability Model
Sa
H vv
an y
U
72 ni
17
1.2 Properties of probability function P
1 3 Independence Us Mutually Exclusive
1 4 Conditional Probability
2 Uni variate Random Variable Review 2.1 v2.4
2 1 Definition
Classification
2.2 Cumulative Distribution Function
2.3 Probability Function
2 4 Expectation
CDF
pf pmf pdf
Variance
2.5 Moment Generating Function MGF
3 I
3.7 RADYR Chap3423
19091104661
Lect
Lec 4
Part I
Probability Review
1 1 Probability Model
all possible
Sample space s
outcomesfiomatrindom
an outcome or a set
Event A B
Probability function p A satisfying
O E PLA E I
a
experiment
i.e subsetof S
of outcomes
45442 A 12,4 6
A
Y A ES
Sa
H vv
an y
U
72 ni
17
b Pls
If Al Az
C
then
ES are mutually exclusive Ai nAj 0
PIEAi ÉPCAi
T probability model
Mt 3 ERGEN
1 2 Properties of probability function P
A PA
I PA
P AUB P A
I PS
PLO
0
PCA PCB PCC PCANB
pcanc p Bnc PLAn BAC
P B PLANB PCAUBUC
If A EB thenPLA s PCB
C
PlanB
pa
Bi s
0
E
M
BB
piaBiB5
IFPI
Ppf
plan B
e DeMorgan's Law
ATB A n B KEEEtta I FETE e g FABRE AUBuc
An Bnc
1.3 Independence vs Mutually Exclusive
Independence
A and B are independent if
Mutually Exclusive A and B are mutually
independence
i
PA
A
O
AND UE
f
PB
FEEFREIGHT
P ARB
exclusive
if PLANB
mutually exclusive
0
A
B independent rt
B mutuallyexclusive
independent
19091104661
PCAPCB
o
BATHE
ME
A B KAI mutuallyexclusive
PCA o or p B
o ortop o
Exercise 1 Given 3 events A B and C We know PLA
pC
P B 0.4
0.5
If A
I
o3
B C are mutually independent try to find Plan B ni
10 PCFn BAE
PL A U B UC
I
PLAUBUC
plates pineifeng.gg
pya pass peg pianist
PLAPCB P E 0.7 0.6 0.5
I
2 PLANBAT
Sa
H vv
an y
U
72 ni
17
A B Cmutuallyindependent IÉBECE
A B
A B
A HB
PLANB p B PCANB P B PCAPCB PCB CI pCAD PLBI PCA
If B C andA C are independent while A B are mutuallyexclusive
2
find PC E V BUC
i PCEu Bud
PCA
2
pcan B
MYtpcanbnd
PCEIEMIIE.mg
PIE Pip
ig
Taped
p
pin ubuy pcave Boo
P
End pogB
pgqfffEpfftpancr.ua
I PCA PCAnBUC IPSEEBLY Anc
I P A plan c
3
If A B andA C are independent while B C are mutuallyexclusive
Bnc 0
BT s
PLA nEBU
find
EI
pCA n IB UE
pl A n
PLA
Bye
PLANS
Pingry
19091104661
P A 0.3
PCA
o3
1 4 Conditional Probability
Definition A and B are events defined on S PCA B
Product Rule PLANB
Independence
A
B
PCA B PCB
if PCA E
PLANE PCA PLANB
É Plan Bi
PLA
or PCB A
PLANB
Bin Bj p
PLANB
I Bi S
EX PLANB PLA PCB
PCA B
pA
PCB
KEF ttiiE5bbik
IELAIBi.pe
AIB
PA
Sa
H vv
an y
U
72 ni
17
pA
P Ap
or pCBIA
19091104661
pB
PCB
0
Exercise 2 84122247covid 19HEALYSEEK TELLABNER
KRIEFEFFETERIFFE MAMA 13171467 2254 95
TEENEFINALETA I 1 AFFINITY
99
1
t's 1
1
HI 99 1
Sa
H vv
an y
U
72 ni
17
ÉÉ
Plea
1
999 x 5
98
PLBR n ITIS
PIPE n Hm
P BA 7FED PIT FI
PC1317 FB pt BI
t 5
98 x 1
2 ARETE TERETEFIERY
P THE
1312
x 99
BEAL Ania ANDFIFA ETR EEE
PCFIFABR
p BA
PCPR FB PCH
P B12
1
98
98
1
19091104661
5
997
IT HEHEtFEEL4744BEYELEAR 8412ITEM 309E
H54Egenertmath mÉÉÉactsc 742 stat 812 cs 4thfarm
Exercise 3 ZR
BI
IT
JimT544KEEPHER C An FEEDMY17BAKIA 5KYETtoBe
I 5424
EFFECTBYEBYEgeneralmath
Sa
H vv
an y
U
72 ni
17
E
a IETFgeneralmathMy17ESTEEM
1
3
E
5
HEHE generalmath 17EFREM
P LEKTA T GM
I
I
4
HE A 4213EYEYEH
E E I
E Io 4
5
5
x
19091104661
PLIKAGM
l
E
É
E
3 57 Y
i 54
Part I Uni variate Random Variable Review
Classification
2 1 Definition
Definition A random variable is a function from S to R
P X
x
is defined for all x EIR
Continuous
Discrete
countable
Classification
St
interval
Sa
H vv
an y
U
72 ni
17
2 2 Cumulative Distribution Function
Definition
Fix
i
P xx
for alexei
non decreasing
ii ftp.Fcxs 0
e.g Fix
Fix
fi nFix7
right
9ETAHHiii
Properties
continuous
9 Effy
o
to
net
timafix Fia
TEAM A function t.TT AnLE34E4FEx valid CDF
cont
continuous
cont
its riv
right
discrete r v
Ftp
left
Gien DENETITLE rightside rightcont
x
19091104661
Exercise 4 Verify if Fix
10
20
x so
THE
If
x
4
o
0
a
t i Fix Fix
1
Sa
H vv
an y
U
72 ni
17
A cts r v has CDF as follow Fox
Find all consttsCLed Pa
continuous riv
Flo
Ci e
7 1
Fu
Cz Cz e
props
IIIs
IMF x
C
cottage Enemy
0
x
I
a
i
non decreasing
right continuous
continuous
y
Exercise 5
valid CDF
hefty
Fix
30
is
É
Cz
l
i
Cz
i
l
Cz 1 e
03 1
19091104661
t
i
4 1
X o
c e
Ie
x
a
Cu
C
Cz Cz e t
Xz
I
Exercise 6
Suppose
X is a r v with CDF
Xe I
O
F
yea
zig X 21
i
I Find PC E
XE
FL E
FIE
PC ECXEI
2 to
2
I
I
ttft3
3
Sa
H vv
an y
U
72 ni
17
8 4
d
find peat
PLY 4
PIXEL PLX H
FCF Lim FIX
FL
FLI O
jumpbyaFK
3 Find PIX 1 and PIX 1
PLX
1
FL D
PIX 1
FLD
4 Find P 2
43
P Z X
5
3
Pete x so
YI Fix
GI Fix
713
24
3
1
É
I
F 2
I
0
I
0
I PY
D
p Yeo
pg
F IO
FL D PIX D
21
12.4ft I
19091104661
4
PLA 1
2.3 Probability Function
fix
Definition
Pix x
X discrete
pmf
X continuous pdf
Fix
P
I o k
plz pyffi fix 70
e.g fix ke x
ii Eat1 7 1 orInfix dx 191742 EEpft mfxkp.tk
properties
e g tix
fix
LEx
A co Tx
fix Ig
A
o
Exeter
m
Intinious
any H function t.tt B4LFai.iiFEj Fxvalidpf
Hamma Function Tik
If
Figg's
Sa
H vv
an y
U
72 ni
17
x
A
iii
so
a
T1
fixed
2
PCE JI
1
b T K CK D TIK D
integration
dx 513
K I
It cK 2
egg egg
CK 1
gammafunction th THEN
by parts
HEHEHE CODEÉFÉÉ
EhrCDF Fipf
a discrete
fix P Xx
pyx
Fix
b continuous fix
FIER
FIFER
l
ox
P Xcx
Fix limpet
It
I
I fast
fi CDF
a discrete FIX P XEN EEP Kt
it It
b continuous Fix P Xe x
EEPpf
IF
19091104661
Éfit
fatso
2
Exercise 7
fix
c x2 e x
o ex
find value of c
x
Also find
CDF of X
x2 e dx
Jc
c
i
Ix
XE O
2
x
e
dx
C 713
FIX
fix
o
FIX
P IX ex
c 2
1
i c
t
LIE
dt
o
C 2
P YEE
Sa
H vv
an y
U
72 ni
17
l
0
It
fit
L
L It
e
tdt
III e tdtfudv uv
th.dr
fud u
e
tdztdt.ve
du
e
t
IT Eet tafféttdt
If x'e zffte
tdttdtu
t.de
du dt
V
e
e
t
ÉEXéYzl té4 fÉtdt
E x'e 2 Get l e
t
ze
Xt
axe
Xyz
IE
x
to
no
o
Fox
LE
IEEE
É
e
0
0
19091104661
73
12 2
Fix
t
X 0
2
1
2.4 Expectation
Variance
Effigy
Definition
Expectation
Notes
t
X discrete
dx
fix
IF
X continuous
A
ftp.onfygxitEgghenEllxt
i
it i
L
Sa
H vv
an y
U
72 ni
17
I gix fix dx
m
enemy
Definition Van X
EX
Variance
property
EL X END
Var X
EX
VartaxPgffar
19091104661
t
x continuous
E x
Ex
We have pdf for X
Exercise 8
Find E X
Varix
EX
S
If
fix
Lec2 Pl
X
Biff
a
xadx
dx
Bel tnx I
p.ae 5x Bdx.i.p
p.ae ftp.xtmt
p 2B Ip o at B
B
I
as 1
to
Exata
BYE
Sa
H vv
an y
U
72 ni
17
B 2B 2
ixa
By
p 1
Varix
Ey
IIIs
1
I
dx
tnx
xad
atxatil
only existswhenp t
p z
PT B dxpya.enxtIse
p ti p z
2P
Ddx
5x
p
i op Do
x Pt
affect
p
ft
p 2B Hi o at
Ptl
x
B
varix
ELIEL
270
x
x
Bal
c at
FI
II
JI
19091104661
sp a
pay
rapt p z
Exercise 9 Let X Bin in p
fix
EC D
I p CI p
fix
X
X
o I
i
i
n
Éx
n
ftp.xittp
x
II
Eatm l
Sa
H vv
an y
U
72 ni
17
I
try to find ELD
I
iii
In
x
p cap
iii
n x
É attilio
É II p lips
z p cap
n
n
É
up
my
pI cap
n x
let y x i i x ytl
n yet n y I Ch D y
ch D y
an
Eilislpinift
pt I
up I
up
pfI
t
19091104661
Practice I
X
Var X
find Var X
Bin chip
ELX
EX
E Lx
E XIX I
X
EEXCH
ELD E X
EI XIX 1
É
Xt
xcx i
H p cap MY
I p is p
Sa
H vv
an y
U
72 ni
17
FIND Ida
2
In x P cap
ex
Is at It
n in 1
let y x
É
II
p
non 1
E3 ny p't
non 1
É http
Atp np tap nip
np np
n.cn i
yt2
p't up n'p
npoi p
19091104661
t
Lp
x
n in 1 p
Var x
n t
p ctp
2
non op
n
p cap
n x
n x
i
pin as y
pf ztixs
Exercise 10
I
T
Ect
0
I
I
Van 2
I
FEELY
95
5
fix
ELY
3
Var 2
1
t
1
Sa
H vv
an y
U
72 ni
17
PIX
fly
and
my mom
tfX.tze xdx
I 712
I Jtx e x dx
fi
EL max X 5
discrete
x
i
Let Y Max X 5
2
find E x
Nco
X 5
discrete
giggle
Sty fly dy
x fix dx
5 PIX 5
It
continuous
5 f 15
5
I
5
te
1
4 Var X
Édx
5
ELA
Var X
E X
EIN
YE Ix teddy
If
I 173
i
Varix
i
Vortex 11
dx
X e
I
I
2
I
4 Varix
3
19091104661
x.zetdx byparts
2 5 MomentGeneratingFunction
Notes tTHEN
EERIEETIMGFTITI BYFIVE IN
I Mo
EsteMii't
Properties
ii
iii
iii E X
to
d
ME leo
MY o
E x2 M 10
M
MY o
Sa
H vv
an y
U
72 ni
17
L
É
BENI TY
VarX
o
ECetY
Transformation
of MGF Y ax b Myit Eleggy
19091104661
laxtbs
E et
ebtMxcat
ate th h
19091104661
Sa
H vv
an y
U
72 ni
17
19091104661
Sa
H vv
an y
U
72 ni
17
19091104661
Sa
H vv
an y
U
72 ni
17
19091104661
Sa
H vv
an y
U
72 ni
17