Appunti completi di Teoria dei segnali

Y/X

1 − 2

2

(,

) = Y/X

Y/X √2

Y/X

= = = =

= = 0; = =

=

Y/X

{ 2 2

(1

= − )

Y/X

= 0.9, = 1 =2

|

= = 1.8

Y/X =2

{ 2 2

(1

= − ) = 0.19 ⇒ = 0.44

Y/X

Y/X [0.6,

3] 1.8

= 0.99 |

= = 1.98

Y/X =2

{ 2 2

(1

= − ) = 0.02 ⇒ = 0.14

Y/X

Y/X 15

2.2 Variabili aleatorie bidimensionali

=0 |

= = 0

Y/X =2

{ 2 2

(1

= − ) = 1 ⇒ = 1

Y/X

Y/X

()

()

(⋅) 16

2.2 Variabili aleatorie bidimensionali

= ()

()

= { ≤ } = {() ≤ }

() −1 −1

() ()} [ ()]

= { ≤ } = {() ≤ } = { ≤ =

[()] ()

=

2

−

() 2

= ()

2

2

2 2 2

− − −

() () 2 2 2

= ∫ = ∫ = = 1 − >0

[ ]

2 2 2

2

0 0 0

(0)

= 0 < 0

2 ()

−

[()] ()

2

= 1 − = =

2

2 ()

− 2

=1−

2

2 ()

− = ln(1 − )

2

2 2

√−2

() = ln(1 − ) [0,1]

∈

()

−

()

= ()

= ()

[()] ()

=

17

2.2 Variabili aleatorie bidimensionali

− −

() ()

= ∫ = = 1 − , >0

[− ]

0

0 −()

1 − =

−() = ln(1 − )

1

() = − ln(1 − )

= = 0

= = 1

=0 2 2

1 +

−

(, ) 2

= 2

2

2

=

()

= { ≤ } = ≤ }

{

{(,

≡ ): ≤ }

>0 ≤ 18

2.2 Variabili aleatorie bidimensionali

<0 ≥

0 ∞ ∞

() (, (, (,

= ∫ ∫ ) = ∫ ∫ ) + ∫ ∫ )

−∞ 0 −∞

0 ∞

()

() (, (,

= = ∫ − ) + ∫ )

−∞ 0

∞ || (,

= ∫ )

−∞ (, ()

)

= ∞ ∞

2 2 2 2

1 2

+ +

− −

() || 2 2

= ∫ = ∫

( )

2 2

2 2

2 2

−∞ 0

∞

2 2

(1+ )

1 1 1 1

− 2

= =

[ ]

2

2 2

1 + 1 +

0

= 0

∞ 1 1

(2) ()

2 2

= − = = ∫ = ∞

2

1 +

−∞ 19

2.2 Variabili aleatorie bidimensionali

=+ ,

(, () ()

) =

=+

()

= { ≤ } = { + ≤ } = { ≤ − }

(

= − )

() (

= − )

=+

(, () ()

) =

20

2.2 Variabili aleatorie bidimensionali

(z/x) (

= − )

Z/X Y/X ()

(Z/X) (,

= )

Z/X

∞ ∞ ∞

() (, ( () (,

= ∫ ) = ∫ − ) = ∫ − )

Y/X

−∞ −∞ −∞

∞

() () (

= ∫ − )

−∞

=

{ ≤ } > 0

()

= { ≤ } = { ≤ } = {

{ ≥ } < 0

1

> 0

( )

()

= {

1

− < 0

( )

=

1

(z/x)

= ( )

Z/X Y/X

||

1 1

(, (z/x) () ()

) = = =

( ) (, )

Z/X Y/X

|| ||

∞ ∞

1 1

() ()

= ∫ = ∫

(, ) ( )

|| ||

−∞ −∞ 21

2.2 Variabili aleatorie bidimensionali

√ 2 2

= +

(, ) ()

{√ 2 2

()

= { ≤ } = + ≤ }

√ 2 2

≡ {(, ): + ≤ }

= cos

{ = sin 2

() (, (

= ∫ ∫ ) = ∫ ∫ cos , sin )

0 0

2

()

() (

= = ∫ cos , sin )

0

, = = 0 = = =1

2 2

1 +

−

(, 2

) = 2

2

2

2 2 2

()

1

−

() ( 2

= = ∫ cos , sin ) = ∫ 2

2

2

0 0

2 2

− −

2 2

= 2 =

2 2

2 2

2

0 (, )

0 22

2.2 Variabili aleatorie bidimensionali 2

0 −

√ 2 2 } ( ) 2

{} = { + ≤ } = { ≤ = = ∫

2

0 0 0 2

0

2

2

0

− −

2 2

= =1−

[− ]

2 2

, , … ,

1 2

( ) }

, , … , = { ≤ , … , ≤

, ,…, 1 2 1 1

1 2 ( )

, , … ,

, ,…, 1 2

1 2

( )

, … , =

, ,…, 1 …

1 2 1 2

1

2

= ( )

…

1

2

= ( )

…

23

2.2 Variabili aleatorie bidimensionali

⋯

11 1

⋮ ⋱ ⋮

= ( )

⋯

1

= ( , = {( − −

) ) ( )}

1

2

=

=

1 1 −1

−

2(− ) (− )

=

()

(2) √| |

2

• − 1×

( )

• −1

×

• − ×1

( )

=

=2

1 1

= =

( ) ( )

2 2

2

12

1

= ( )

2

21

2

= = {( − −

)( )}

12 21 1 2

1 2

12

=

1 2

2

1 2

1

= ( )

2

1 2 2 24

2.2 Variabili aleatorie bidimensionali

2 2 2 2 2 2

2 2

| | (1 )

= − = −

1 2 1 2 1 2

2 2 2 2

− −

1 1

−1 2 1 2 2 1 2

= )=

( ( )

2 2 2 2 2 2

2

| | (1 )

− − −

1 2 1 1 2 1

1 2

1

−

2

1

1 2

1

= 1

2

1 − − 2

( )

1 2 2

1 1

=

2

√(1 )

2 −

(2) √| |

2

1 2

1

−

2

−

1

1 2 1

1

− , −

( ) ( )