Download - Chapter4 Continous Distribution
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8/18/2019 Chapter4 Continous Distribution
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Continuous Probability
Distributions
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Continuous Random Variables and
Probability Distributions
• Random Variable: Y
• Cumulative Distribution Function (CDF): F (y )=P(Y ≤y )
• Probability Density Function (pdf): f (y )=dF (y )dy
• Rules !overnin! continuous distributions:
f (y ) " # ∀ y
P(a≤Y ≤b) = F (b)$F (a) =
P(Y =a) = # ∀ a
∫ b
ady y f )(
1)( =∫ ∞
∞−dy y f
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%&pected Values of Continuous RVs
[ ]
[ ]
( )[ ] [ ]
[ ]
[ ] ( )[ ] ( )
σ σ
σ µ µ
µ
µ µ
µ µ µ µ
µ µ µ µ
µ σ
µ
a
aY V ady y f yady y f aay
dy y f babaybaY E baY E baY V
baba
dy y f bdy y yf ady y f baybaY E
Y E Y E
dy y f dy y yf dy y f ydy y f y y
dy y f yY E Y E Y V
dy y f y g Y g E
dy y yf Y E
baY =
==−=−=
=+−+=+−+=++=+=
=+=+=+
−=+−=
=+−=+−=
=−=−==
===
+
∞
∞−
∞
∞−
∞
∞−
∞
∞−
∞
∞−
∞
∞−
∞∞−
∞∞−
∞∞−
∞∞−
∞
∞−
∞
∞−
∞
∞−
∫ ∫ ∫
∫ ∫ ∫
∫ ∫ ∫ ∫
∫ ∫
∫
222222
22
2222
2222
222
)()()()()(
)()()()()(
)1()(
)()()()(
)1()(2
)()(2)()(2
)()())(()(:Variance
)()()(
e)convergencabsolute(assuming )()(:ValueExpected
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'niform Distribution
• 'sed to model random variables tat tend to occurevenly* over a ran!e of values
• Probability of any interval of values proportional to its+idt
• 'sed to !enerate (simulate) random variables fromvirtually any distribution
• 'sed as non$informative prior* in many ,ayesiananalyses
≤≤−=
elsewhere 0
1
)(
b yaab
y f
>
≤≤−−
<
=
b y
b yaab
a y
a y
y F
1
0
)(
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'niform Distribution $ %&pectations
( )
( ) [ ]
)(28870
1212
)(
12
)(
12
2
12
)2(!)("
2!
)()()(
!
)(
)(!
))((
)(!!
11
2)(2))((
)(2211)(
2
2222222
22222
22
22!!!22
222
ababab
ababbaabababba
ababbaY E Y E Y V
abba
ab
abbaab
ab
ab y
abdy
ab yY E
abab
abababab y
abdy
ab yY E
b
a
b
a
b
a
b
a
−≈−
=−
=⇒
−=
−+=
++−++=
=
+−++
=−=⇒
++=
=−
++−=
−−
=
−=
−=
+=− +−=−−= −= −=
∫
∫
σ
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%&ponential Distribution
• Ri!t$-.e+ed distribution +it ma&imum at y =#
• Random variable can only ta.e on positive values
• 'sed to model inter$arrival timesdistances for a
Poisson process
>
=
−
elsewhere0
01
)(
# ye
y f
y θ
θ
( ) 011
1
11)( 0
00>−=−−−=
−
== −−−−−∫ yeeeedt e y F y y y
t t y
θ θ θ θ
θ θ θ
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%&ponential Density Functions (pdf)
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%&ponential Cumulative Distribution Functions (CDF)
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/amma Function
)()(
:$etting :integralhe%onsider t
)&1()(integer'anisi that ote
*ropert+)(,ecursive )()0(0
)1(
:*arts b+g-ntegratin )1(
)(
0
1
0
1
0
1
01
0
1
0
1
00
1
0
0
1
α β β β β
β β
β
α α α
α α α
α α
α
α
α
α α α α β α
β α
α
α α α
α α
α
α
Γ ===⇒
=⇒=⇒=
−=Γ
Γ =+−−−=
=+−=−==+Γ ⇒
−=⇒=
=⇒==+Γ
=Γ
∫ ∫ ∫
∫
∫ ∫ ∫ ∫
∫
∫
∞ −−∞ −−∞ −−
∞−−
∞ −−
∞ −−∞−∞ −
−−
−
∞−
∞ −−
dxe xdxe xdye y
dxdy x y
y xdye y
dye y
dye ye yvduuvdye y
evdyedv
dy ydu yu
dye y
dye y
x x y
y
y
y y y
y y
y
y
%0C%1 Function: =%0P(/233214(α))
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%&ponential Distribution $ %&pectations
( )
( ) [ ]θ σ
θ θ θ
θ θ θ θ
θ θ
θ θ θ θ
θ θ
θ θ θ
θ θ θ
=⇒=−=−=⇒
=−=Γ=
===
=
=−=Γ=
===
=
∫ ∫ ∫
∫ ∫ ∫
∞ −−∞ −∞ −
∞ −−∞ −∞ −
22222
22!
0
1!
0
2
0
22
2
0
12
00
)(2)()(
2)&1!()!(1
11
)&12()2(1
11)(
Y E Y E Y V
dye ydye ydye yY E
dye ydy yedye yY E
y y y
y y y
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%&ponential Distribution $ 3/F
( )
[ ] 2222
!2!
22
1
..
0
.
.
.
0
.
0
1
0
1
0
2)0(/)0(//)(
)0(/)(
)1(2)()1(2)(//
)1()()1(1)(/
)1(1
1)10(111)(
1 where
11
11
)(
θ θ θ
θ
θ θ θ θ θ
θ θ θ θ
θ θ θ
θ θ θ
θ θ
θ
θ θ
θ θ
θ θ
θ
θ θ
θ
θ θ
=−=−=⇒
==⇒
−=−−−=
−=−−−=
−=−
==−−=
−=⇒
−===
=
==
−−
−−
−∞−
∞ −∞
−−
∞
−−∞ −
∫ ∫ ∫ ∫
M M Y V
M Y E
t t t M
t t t M
t t
et M
t dyedye
dyedyeee E t M
y
y
t y
t y ytytY
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%&ponentialPoisson Connection
•Consider a Poisson process +it random variable X bein!te number of occurences of an event in a fi&ed timespace
X (t)5Poisson(λt)• 1et Y be te distance in timespace bet+een t+o suc
events
• 6en if Y 7 y, no events ave occurred in te space of y
λ θ θ λ
λ λ λ
θ
1meanwithlExponentiaare*rocess*oissonindistancesarrivals-nter 1
&0)()0)(( :+*robabilit*oisson
)( :urvivallExponentia
0
==⇒
===
=>
−
−
−
y
y
y
e ye y X P
e yY P
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/amma Distribution
•Family of Ri!t$-.e+ed Distributions• Random Variable can ta.e on positive values only
• 'sed to model many biolo!ical and economic caracteristics
• Can ta.e on many different sapes to matc empirical data
>>Γ
=
−−
otherwise 0
0''0)(
1
)(
1 β α β α
β α
α ye y
y f
y
Obtaining Probabilities in EXCEL:
To obtain: F(y)=P(Y≤y) Use Function: =G!!"#$T(y% β
%&)
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/amma%&ponential Densities (pdf)
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/amma Distribution $ %&pectations
( )
( ) [ ]
α β σ
αβ β α αβ β α αβ αβ α
αβ α α
β α α α
α
β α α α
β α β α
β α β α
β α β α
αβ
α
β α α
α
β α β α
β α β α
β α β α
α
α
β α
α
β α
α
β α
α
α
α
β α
α
β α
α
β α
α
=⇒
=−+=−+=−=⇒
+=Γ
Γ+=
Γ+Γ+
=
=Γ
+Γ=+Γ
Γ=
Γ=
=Γ
=
Γ
=
=Γ
Γ=
=Γ
+Γ=+Γ
Γ=
Γ=
=Γ=
Γ=
+∞ −−+
∞ −+∞ −−
+∞ −−+
∞ −∞ −−
∫
∫ ∫
∫ ∫ ∫
222222222
22
22
0
1)2(
0
1
0
122
1
0
1)1(
00
1
)()1()()(
)1()(
)()1(
)(
)1()1(
)(
)2()2(
)(
1
)(
1
)(
1
)(
1
)(
)(
)(
)1()1(
)(
1
)(
1
)(
1
)(
1)(
Y E Y E Y V
dye y
dye ydye y yY E
dye y
dye ydye y yY E
y
y y
y
y y
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/amma Distribution $ 3/F
( )
( )
[ ]2222
222
11
.
.
0
.1
0
1
1
0
1
1
0
1
)()1()0(/)0(//)(
)0(/)(
)1()1()()1()1()(//
)1()()1()(/
)1()()(
1)(
1 where
)(
1
)(
1
)(
1
)(
1)(
αβ αβ β α α
αβ
β β α α β β αβ α
β αβ β β α
β β α β α
β
β β
β α
β α β α
β α
α α
α α
α α
α
β α
α
β
β
α
α
β α
α
β α
α
=−+=−=⇒
==⇒
−+=−−−−= −=−−−=
−=ΓΓ
=⇒
−=
Γ=
Γ=
Γ=
=
Γ
==
−−−−
−−−−
−
∞ −−
∞
−−
−∞
−−
−
∞ −−
∫
∫ ∫
∫
M M Y V
M Y E
t t t M
t t t M
t t M
t dye y
dye ydye y
dye yee E t M
y
t yt y
ytytY
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/amma Distribution 8 -pecial Cases
• %&ponential Distribution 8 α=1
• Ci$-9uare Distribution 8 α=ν/2, β=2 ( ν inte!er)
8 %(;)= ν V(;)=
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4ormal (/aussian) Distribution
• ,ell$saped distribution +it tendency for individuals toclump around te !roup medianmean
• 'sed to model many biolo!ical penomena
• 3any esti'ators ave appro&imate normal samplin!
distributions (see Central 1imit 6eorem)
0''2
1
)(
2
2)(
2
1
2 >∞
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4ormal Distribution 8 Density Functions (pdf)
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4ormal Distribution 8 4ormali>in! Constant
( )
( ) ( )
222
2
2
0
2
0
2
0
2
00
2
1
222
0 0
2
12
0 0
sincos2
1
212
12
2121
212
1
22
12
2
222
)(
2
)(
222
2))1(0(
)1sin(cos
and )2'04)''0(:domains withsin'cos
:5rdinates%o*olarto%hanging
1 :variables%hanging
)orsolvewant to(we :integralhe%onsider t
2
222222
21
2
2
2
1
22
21
22
2
2
2
2
πσ πσ π σ
π θ θ θ θ
θ θ θ θ
σ
θ π θ θ θ
σ
σ σ
σ σ σ
µ
π π π π
π π θ θ
σ
µ
σ
µ
=⇒=⇒=
⇒
===−−−=−=
=+===
=∈∞∈==
== ⇒
=⇒==⇒
=⇒=⇒−=
=
∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫
∫ ∫ ∫ ∫
∫ ∫ ∫
∫
∞
=
−
∞ −∞ +−∞
∞−
∞
∞−
+−
∞
∞−
∞
∞−
+−∞
∞−
−∞
∞−
−
∞
∞−
−∞
∞−
−∞
∞−
−−
∞
∞−
−−
k k k
d d d e
rdrd erdrd edz dz ek
rdrd dz dz r r z r z
dz dz edz edz ek
dz ek
dz edyek
dz dydy
dz y z
k k dye
r
r
r r z z
z z z z
z z y
y
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?btainin! Value of Γ(1/2)
( )
π π
π
π
=
==
=
=
=
Γ ⇒
=⇒=
= Γ
=⇒
=
∫
∫ ∫
∫ ∫
∫ ∫
∞ −
∞ −∞ −−
∞
−−
∞
−−
∞ −
∞
∞−
−
2
2
122
12
22
1
2 :Variables%hanging
2
1 :%onsider ow'
22
12:getweslide'*revious6rom
0
2
0
2
0
2212
2
0
21
0
121
0
2
2
2
22
2
2
dz e
zdz e z
zdz e z
zdz du z
u
dueudueu
dz e
dz e
z
z z
uu
z
z
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4ormal Distribution $ %&pectations
( )
( )
( ) [ ]( )
222
2
222
2!
2#!2!0
212!
0
2
0
2
1
2
1
0
2
2
2
1
0
22
1
22
2
1
2
1
2
1
2
1
)1()()()(
)0()()()(
then'3- : ote
1101)()(
122
2221
21
212
2!
21
21
2
1
2
2
2
2
2
12
2
1 2 :Variables%hanging
2
12
2
1
0)0(02
1
2
1
2
1)(
2
1)()1'0(3
22
22
222
2
σ σ σ σ µ
µ σ µ σ µ σ µ
σ µ σ µ
σ
π π
π π π
π π π
π π
π π π
π
===+=⇒
=+=+=+=⇒+=
=⇒=−=−=⇒
== Γ = Γ ==
=
==
⇒
=⇒=⇒=
=
=
=−−−=
−==
=
=⇒
∫
∫ ∫ ∫
∫ ∫
∫ ∫
∞−−
∞ −∞ −−∞
−∞∞
∞−
−
∞
∞−
−−∞
∞−
∞
∞−
−
−
Z V Z V Y V
Z E Z E Y E
Z Y N Y
Z E Z E Z V
dueu
dueu zdz zedz e z
zdz du zdz du z u
dz e z dz e z Z E
edz e z dz e z Z E
e z f N Z
u
u z z
z z
z z z
z
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4ormal Distribution $ 3/F( ) ( )
( )
( ) ( )
( ) ( )
[ ] ( )
( ) [ ]
( )
( )
+=
+
=⇒
+
+−−
+
=
=
+
+
+−−=
=
++++−++−=
=
+
++
−−+
+−=⇒
++=+
−++−=
=
+−+−=
−
−==
∫
∫
∫
∫
∫
∫ ∫
∞
∞−
∞
∞−
∞
∞−
∞
∞−
∞
∞−
∞
∞−
∞
∞−
2
exp
2
2exp)(
'3 :,Vnormalao densit+over thegintegratinisitsince1' beingintegrallasthe
)(
2
1exp
2
1
2
2exp
2
2)(
2
1exp
2
1
2
2
2
2)(
2exp
2
1
2
2
2
2
2
)(
2exp
2
1)(
2)( :suarethe%ompleting
2
)(2
exp2
1
22exp
2
1
2
1exp
2
1)(
22
2
222
22
2
22
22
222
2
222
2
22
2
2
222
2
2222
2
2
2
2
2
2
222
2
222
2
2
2
2
2
2
2
222222
2
2
2
2
2
2
2
2
2
22
2
22
2
2
σ µ
σ
σ σ µ
σ σ µ
σ
σ µ
πσ σ
σ σ µ
σ
σ σ µ
σ
σ µ
πσ
σ
σ σ µ
σ
σ σ µ µ
σ
σ µ
σ πσ
σ
σ σ µ
σ
σ σ µ
σ
µ
σ
σ µ
σ πσ
σ σ µ µ σ µ
σ µ
σ σ µ
σ πσ
σ
µ
σ
µ
σ πσ σ
µ
πσ
t t
t t t M
t N Y
dyt yt t
dyt t t y
dyt t t t t y y
dyt t t t t y y
t M
t t t
dyt y y
dyty y y
dy y
ee E t M tytY
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4ormal(#@) 8 Distribution of A<
( )
2
1
2
2121
21
0
21
21
2
1
0
21
2
2
0
21
2
0
2
21
21
2
3
)21()21(2
2
21
2
2
1
2
1
2
1
2
12)(
2
1 and :Variables%hanging
2
2
2
2
1
2
1)(
2
22
2222
2
χ
π
π
π
π π
π π
π π
Z
t t t
dueuduu
et M
duu
dz u z z u
dz edz e
dz edz eee E t M
t u
t u
Z
t z
t z
t z z tz tZ
Z
⇒
−=−=
−
Γ=
===⇒
==⇒=
==
==
==
−−
∞
−−−∞
−−
∞
−−∞
−−
∞
∞−
−−∞
∞−
−
∫ ∫
∫ ∫
∫ ∫
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,eta Distribution
• 'sed to model probabilities (can be !enerali>ed toany finite@ positive ran!e)
• Parameters allo+ a +ide ran!e of sapes to model
empirical data
>
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,eta Density Functions (pdf)
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Beibull Distribution
( )
( )
( ) [ ]
+Γ −
+Γ =−=⇒
+Γ ===
−=⇒
+Γ ===
−=⇒
==⇒=
−=
>
−=
−−−==
>>
−−
≤=
∫ ∫ ∫
∫ ∫ ∫
∫
∞ −∞ −∞ −
∞ −∞ −∞ −
−
∞ −
−−
β β α
β α α α
α α
β
β α α α
α α
β
α α
β
α
α α
β
α α
β
α α
β
β α α
β
β β β β β
β
β β β β β
β
β β β
β β
β β
β β
β
11
21)()(
21)(exp
11)(exp)(
)( and :variables%hanging
exp)(
0orexpexp)(
)(
)0'(0exp1
00
)(
2222
2
0
22
0
2
0
122
1
0
11
0
1
0
1
11
0
1
11
Y E Y E Y V
dueudueudy y
y yY E
dueudueudy y
y yY E
u ydy y
du y
u
dy y
y yY E
y y
y y
ydy
ydF y f
y y
y
y F
uu
uu
4ote: 6e %0C%1 function B%,'11(y@α∗,β∗) uses parameteri>ation: α=β, β∗=αβ
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Beibull Density Functions (pdf)
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1o!normal Distribution
( )
( ) ( ) ( )
( ) ( )( ) ( ) ( ) ( )
( ) [ ] ( )22
2
2
2
2222
222
.
.22.2
222
.
.
2.
log
2
1
22
)()(
2
2
)2(exp)2(
2
1)1(exp)1()(
'3)ln( : ote
otherwise 0
0''0
2
1
)(
σ µ σ µ
σ µ
σ µ
σ
µ
σ µ
σ µ
σ µ
σ µ
σ π
++
+
+
−−
−=−=⇒
=
+=====
=
+====
=
>∞
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1o!normal pdfEs