X F ( x ) n X F ( x ) ( n ) X X F ( x ) X X F ( x )=[ F ( x )] n ( n ) X X F ( x ) 1 2 n ( X ,X ,...,X ) ( n ) X

(1)

X

(n)

(X

1

, X

2

, . . . , X

n

) F

X

(x)

X

_(n)

F

X_(n)

(x) = [F

X

(x)]

ⁿ

F

X

(x)

X

_(n)

F

X

(x)

F

_X_(n)

(x) n

(2)

n X

_(n)

(X

1

, X

2

, . . . , X

n

)

F

_X_(n)

(x) = [F

_X

(x)]

ⁿ

=

�

1 − n(1 − F ) + n(n − 1)

2 (1 − F )

²

+

− n(n − 1)(n − 2)(1 − F )

6 + . . .

�

n

n→∞

lim F

_X_(n)

(x) =

�

1 − n(1 − F ) + n

²

2 (1 − F )

²

+

− n

³

6 (1 − F )

³

+ . . .

�

= e

^{−n(1−F )}

n F

X_(n)

(x) = e

−n(1−F (x))

X

(n)

Φ

1

= exp �

−e

^{−α(x−ε)}

�

α > 0 Φ

₂

= exp

�

−

� V x

�

ϑ

�

x ≥ 0, V > 0; ϑ > 0

Φ

3

= exp

�

− � x V

�

ϑ

�

x ≤ 0, V < 0; ϑ > 1

n F

_X

(x) = 1 − e

^−g(x)

g(x) x

g(x) = αx

F

X

(x) = 1 − e

^−αx

,

F

_X_(n)

(x) = e

^−n(1−F^X^(x))

= exp [ −ne

^αx

] = exp �

−e

^{−α(x−ε)}

�

= Φ

₁

(x)

(3)

n = e

^αε

n

F

X

(x) = exp �

−e

^{−α(x−ε)}

�

= exp �

−Λe

^−αx

�

, α > 0

α ε Λ = e

^αε

[0, t]

1/α

f

X

(x) = dF

X

(x)

dx = α exp �

−α(x − ε) − e

^{−α(x−ε)}

�

µ = ε + 0.57722 α σ

²

= π

²

6α

²

�x = ε

α

ε µ σ

ε = µ − 0.450σ 1

α = σ √ 6 π 1/α

σ

α

ε

(4)

σ µ

Cv Ca

Cv = σ

µ = π

√ 6(αε + 0.57722) = π

√ 6(ln Λ + 0.57722) Ca = µ

³

σ

³

= 1.1396

Cv Λ Ca

α ε y = α(x − ε)

F (y) = exp �

−e

^−y

� f (y) = exp �

−e

^−y

− y �

�

y µ(y) σ(y)

� y = 0

µ(y) = γ ∼ = 0.5772 σ(y) = π

√ 6 γ

X

i

T

T = 1 1 − P

ⁱ

P

i

X

i

T

X

T

X

_T

= � − 1 α ln

� ln

� T T − 1

��

= �

� 1 − 1

α� ln

� ln

� T T − 1

��

(5)

α � X

T

X

T

= µ

�

1 − Cv

� 0.45 +

√ 6 π ln

� ln

� T T − 1

��

X

T

µ

K

T

X

T

= µK

T

X

T

K

T

N

(x

i,1

, . . . , x

i,ni

) n

i

ξ

indice,i

N ξ

indice,i

ξ

indice,i

µ

i

�x

i

x ˘

i

K

T

K

T

= x

i,T

ξ

indice,i

, i = 1, . . . , N K

_T

ξ

indice

(6)

µ

i

K

T

K

T

= 1 − Cv

� 0.45 +

√ 6 π ln ln

� T T − 1

��

K

T

Cv

h = h(t)

h(t) = at

ⁿ

h t

mm/ora n

t < 1 t > 1 h = h(t)

h t

t < 1 t > 1

i = i(t)

t t < 1

(7)

0 20 40 60 80 100 120 140

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

h [mm]

t [ore]

at^(n)

h

t

P (h

t

) h

t

(h

t

− t)

P (h

t

)

(8)

t T

h

t,T

= µ

t

K

T

µ

t

t K

T

K

T

= 1 − Cv

� 0.45 +

√ 6 π ln ln

� T T − 1

��

µ

t

t K

T

Cv Cv

µ

_t

= ¯ at

ⁿ

¯

a n a

µ

1h

h

t,T

= ¯ a · K

^T

· t

ⁿ

(9)

¯

a n Cv

µ σ α ε

¯

a n

µ

_t

t Cv

Cv t

α ε K

_T

t = 1h ¯ a n

T h

T

(t) = at

ⁿ

a

T

a = ¯ aK

T

= ¯ a

�

1 − Cv

� 0.45 +

√ 6 π ln ln

� T T − 1

��

t = 1 a µ(a) = ¯ a

K

_T

(10)

� y

20

= α(a

20

− ε) y

500

= α(a

500

− ε)

a

₂₀

a

₅₀₀

a

y

20

y

500

T = 20 T = 500 P (y)

y P (a) = P (y) T

y

T

= − ln [− ln P (a)] , P (a) = 1 − 1 T

α ε

t = 1

 



 

ε = y

500

a

20

− y

20

a

500

y

500

− y

²⁰

α = y

₂₀

a

20

− ε

α ε

a

h

t=1

µ(a) σ(a)

Cv

σ(a) = 1.283 α

¯

a = µ(a) = ε + 0.450σ Cv = σ(a)

µ(a)

K

_T

a

(11)

n

n n n n

n

n n n

n

(T, n

T

)

(12)

0.186 0.187 0.188 0.189 0.19 0.191 0.192 0.193 0.194 0.195 0.196

10 100 1000

n

Tr [anni]

Stazione di Mantova - cod. 758

n

n log(20) log(500)

n

T

= n

20

+ n

500

− n

20

log(500) − log(20) [log(T ) − log(20)]

a

20 500 T = 100 T = 200

a

(13)

|a

^{P AI,T}

− a

^T

|

a

T = 200

t = 1

a

(14)

(15)

0.01 0.050.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99 0.999

10 20 30 40 50

2 5 10 20 50 100 200 500

F(h) Tr [anni]

h(t=1h) [mm]

i/(N+1) Gumbel alfa= 0.23294; eps=19.79416 (i-0.5)/N PAI

0.01 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99 0.999

10 20 30 40 50 60 70 80 90 100

2 5 10 20 50 100 200 500

F(h) Tr [anni]

h(t=1h) [mm]

(16)

0.01 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99 0.999

10 20 30 40 50 60 70 80

2 5 10 20 50 100 200 500

F(h) Tr [anni]

h(t=1h) [mm]

n n

20 500 T = 100, 200

|n

P AI,T

− n

T

|

n T = 100 T = 200

n

(17)

n T = 100

n T = 200