Outline Outline Gumbel Gumbel Asymptotic Distributions - - PowerPoint PPT Presentation

SLIDE 1

1

• G. Ahmadi

ME 529 - Stochastics

• G. Ahmadi

ME 529 - Stochastics

Outline Outline

• Gumbel

Gumbel Asymptotic Distributions Asymptotic Distributions

• Type 1: Largest

Type 1: Largest Gumbel Gumbel Distribution Distribution

• Type 1: Smallest

Type 1: Smallest Gumbel Gumbel Distribution Distribution

• Type 2: Largest

Type 2: Largest Weibull Weibull Distribution Distribution

• Type 3: Smallest Distribution

Type 3: Smallest Distribution

• Simulation of Random Variable With

Simulation of Random Variable With Known Distribution Known Distribution

• G. Ahmadi

ME 529 - Stochastics

( ) { } { } { } { } ( ) ( ) ( )

x F ... x F x F x Y P ... x Y P x Y P x Y P x F

n

Y Y Y n n ,..., , i i X

2 1

2 1 2 1

= ≤ ≤ ≤ = ≤ =

=

3 2 1

Let X be the largest of n independent Let X be the largest of n independent random variables Y random variables Y1

1, Y

, Y2

2, …,

, …, Y Yn

• n. Then

. Then

( ) ( ) ( )

n Y X

x F x F =

( ) ( ) ( ) ( )

x f x F n x f

Y n Y X 1 −

=

If Y If Yi

i are identically distributed:

are identically distributed:

• G. Ahmadi

ME 529 - Stochastics

Type 1: (Largest) Type 1: (Largest) Gumbel Gumbel Distribution Distribution Distribution Distribution Density Density ( )

( )

{ }

u x X

e x F

− −

− =

α

exp

( ) ( )

( )

{ }

u x X

e u x x f

− −

− − − =

α

α α exp

α µ 5772 . + = u

X 2 2 2

6α π σ =

X

+∞ < < ∞ − x

1.5 1.0 0.5

• 0.5

0.0

x fX(x)

u=0.275 α=2.566 1

SLIDE 2

2

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ME 529 - Stochastics

Type 1: (Smallest) Type 1: (Smallest) Gumbel Gumbel Distribution Distribution ( )

( )

{ }

u y Z

e z F

−

− − =

α

exp 1

( ) ( )

( )

{ }

u z Z

e u z z f

−

− − =

α

α α exp

α µ 5772 . − = u

Z 2 2 2

6α π σ =

Z

+∞ < < ∞ − z

Distribution Distribution Density Density

1.5 1.0 0.5

• 0.5

0.0

z fZ(z)

u=0.275 α=2.566 1

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ME 529 - Stochastics

Type 2: (Largest) Type 2: (Largest) Weibull Weibull Distribution Distribution ( )

k

x u X

e x F

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −

=

( )

k

x u k X

e x u u k x f

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − +

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ =

1

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − Γ = k u

X

1 1 µ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − Γ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − Γ = k k u

X

1 1 2 1

2 2 2

σ

≥ x

Distribution Distribution Density Density

1 > k 2 > k

2 4 6

fX(x) x

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ME 529 - Stochastics

Type 3: (Smallest) Type 3: (Smallest) Extremal Extremal Distribution Distribution

( )

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − − − =

k Z

u z z F l l exp 1

( )

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − − =

− k k Z

u z u z u k z f l l l l l exp

1

l ≥ z

Distribution Distribution Density Density

l > z ( )

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + Γ − + = k u

Z

1 1 l l µ

( )

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + Γ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + Γ − = k k u

Z

1 1 2 1

2 2 2

l σ

• G. Ahmadi

ME 529 - Stochastics

Simulation of Random Variables with a Simulation of Random Variables with a Known Distribution Known Distribution F FY

Y(y

(y) )

( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≤ ≤ =

• therwise

u u fU 1 1

( )

U F Y

Y 1 −

=

has the desired distribution function has the desired distribution function F FY

Y(y

(y). ). Given that U is Given that U is uniform random uniform random variable with variable with Then Then

SLIDE 3

3

• G. Ahmadi

ME 529 - Stochastics

Exponential Exponential

( ) ( )

y u e y f

y Y λ

λ − =

λ U Y ln − =

Weibull Weibull

( )

λ U Y − − = 1 ln

( ) (

) ( )

y u e y F

y Y λ −

− = 1

• r
• r

( ) (

) ( )

y u e y F

y Y

β

α −

− = 1

( ) ( )

y u e y y f

y Y

β

α β

αβ

− −

=

1

β

α

1

ln 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛− = U Y

• G. Ahmadi

ME 529 - Stochastics

Gumbel Gumbel

( )

( )

{ }

u y Y

e y F

− −

− =

α

exp

Gaussian Gaussian

[ ]

α U u Y ln ln − − =

2 1 2

2 sin ln 2 U U Y π − =

2 1 1

2 cos ln 2 U U Y π − =

( ) ( )

y u y u y F

k Y

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = exp

( )k

U u Y

1

ln − = ( ) ( )

y u u y y F

k Y

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = exp 1

( ) [ ]k

U u Y

1

1 ln − − =

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ME 529 - Stochastics

Concluding Remarks Concluding Remarks

• Type 1: Smallest

Type 1: Smallest Gumbel Gumbel

• Type 1: Largest

Type 1: Largest Gumbel Gumbel

• Type 2: Largest

Type 2: Largest Weibull Weibull

• Type 3: Smallest

Type 3: Smallest

• Simulation of Random Variable With

Simulation of Random Variable With Known Distribution Known Distribution

• G. Ahmadi

ME 529 - Stochastics