Somma di 2 variabili Normali standard s-indipendenti. W X Y= +

( )

( )

2

2

2

2

1 ; .2

1 ; .2

x

X

y

Y

f x e x

f y e x

π

π

−

−

= −∞ < < ∞

= −∞ < < ∞

( )

2 2

2 21, , ; , .2

x y

X Yf x y e x yπ

⎛ ⎞− +⎜ ⎟⎜ ⎟⎝ ⎠= −∞ < < ∞ −∞ < < ∞

Per calcolare la fdp di W opero cambiamento di variabili da X ,Y a W,U e poi marginalizzo: W X Y X W U

U Y Y U= + = −

⇔= =

1 11 1

0 1

X XW UJY YW U

∂ ∂−∂ ∂= = = =

∂ ∂∂ ∂

( ) ( ) ( )( )( )

( )

2 2

2 2

2 2, ,

2 2

1, , , , 12

12

w u u

W U X Y

w u u

f w u f x w u y w u J e

e

π

π

⎡ ⎤−⎢ ⎥− +⎢ ⎥⎣ ⎦

⎡ ⎤−⎢ ⎥− +⎢ ⎥⎣ ⎦

= ⋅ = ⋅ =

= =

( ) ( )( )2 2

2 21, ,2

w u u

W W Uf w f w u du e duπ

⎡ ⎤−⎢ ⎥− +

+∞ +∞ ⎢ ⎥⎣ ⎦−∞ −∞

= ⋅ = ⋅∫ ∫

La variabile U ha supporto, ( ),−∞ +∞ , che non dipende da W. , , ,w u u w w u w u−∞ < − < +∞ −∞ < < +∞ −∞ < < +∞ −∞ < < +∞⇔ −∞ < < +∞

Per effettuare la marginalizzazione riscriviamo come segue la ( ), ,W Uf w u :

:

( )( )

( )

( )

( )

2 2 2 2 2 2 2

22 2 2

222

2

2 2 22 2 2 2 2

1 1 2 2 22 2 2

2112 1 22 2

12 2

1 1 1, ,2 2 2

2 12 2 2

1 12 2 2 1 2

12 2

w u u w wu u u w wu u

W U

ww w wu u

u wu ww

w

f w u e e e

e e

e e

e

π π π

π π

π π

π

⎡ ⎤− ⎡ ⎤ ⎡ ⎤− + − +⎢ ⎥− + − + −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ ⎤− ⎢ ⎥ ⎡ ⎤− − − +⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

⎡ ⎤− +⎡ ⎤ ⎢ ⎥−− ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡−

= = = =

= ⋅ =⋅

= ⋅ =⋅ ⋅

=⋅ ( )

( )2212 1 21

2 1 2

u w

eπ

⎡ ⎤−⎤ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⋅⋅

Quindi marginalizziamo:

( ) ( )

( )

( )

( )

22

22

2

2112 1 22 2

2112 1 22 2

12 2

1 12 2 2 1 2

1 12 2 2 1 2

12 2

u ww

W

u ww

w

f w e e du

e e du

e

π π

π π

π

⎡ ⎤−⎡ ⎤ ⎢ ⎥−−+∞ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

−∞

⎡ ⎤−⎡ ⎤ ⎢ ⎥−− +∞⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

−∞

⎡ ⎤− ⎢ ⎥

⎢ ⎥⎣ ⎦

= ⋅ ⋅ =⋅ ⋅

= ⋅ ⋅ =⋅

=⋅

∫

∫

Risulta quindi ( )0, 2W N∼ [W=X+Y ha ovviamente supporto ( ),−∞ +∞ ] Nota:

( )

( )2212 1 21 1

2 1 2

u w

e duπ

⎡ ⎤−⎢ ⎥−+∞⎢ ⎥⎣ ⎦

−∞

⋅ ⋅ =∫ .

La funzione integranda è la pdf di una Normale di media 2w e varianza 1

2 1,

2 2wU N⎡ ⎤⎛ ⎞

⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦∼ .

Somma di 2 variabili Normali non-standard s-indipendenti. W X Y= +

( )

( )

2

2

12

12

1 ; .2

1 ; .2

X

X

Y

Y

x

XX

y

YY

f x e x

f y e y

μσ

μσ

πσ

πσ

⎛ ⎞−− ⎜ ⎟

⎝ ⎠

⎛ ⎞−− ⎜ ⎟

⎝ ⎠

= −∞ < < ∞

= −∞ < < ∞

( )

2 2121, , ; , .

2

Y X

Y X

y x

X YY X

f x y e x y

μ μσ σ

πσ σ

⎡ ⎤⎛ ⎞ ⎛ ⎞− −⎢ ⎥− +⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦= −∞ < < ∞ −∞ < < ∞

Per calcolare la fdp di W opero cambiamento di variabili da X ,Y a W,U e poi marginalizzo: W X Y X W Z

Z Y Y Z= + = −

⇔= =

1 11 1

0 1

X XW ZJY YW Z

∂ ∂−∂ ∂= = = =

∂ ∂∂ ∂

( ) ( ) ( )( )2 2

, ,1 1, , , , exp 1

2 2X Y

W Z X YY X X Y

w z zf w z f x w z y w z J μ μπσ σ σ σ

⎧ ⎫⎡ ⎤⎛ ⎞ ⎛ ⎞− − −⎪ ⎪⎢ ⎥= ⋅ = − + ⋅⎨ ⎬⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭

( ) ( )2 2

1 1, , exp2 2

X YW W Z

X Y X Y

w z zf w f w z dz dzμ μπσ σ σ σ

+∞ +∞

−∞ −∞

⎧ ⎫⎡ ⎤⎛ ⎞ ⎛ ⎞− − −⎪ ⎪⎢ ⎥= ⋅ = − + ⋅⎨ ⎬⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭

∫ ∫

La variabile Z ha supporto, ( ),−∞ +∞ , che non dipende da W. , , ,w z z w w z w z−∞ < − < +∞ −∞ < < +∞ −∞ < < +∞ −∞ < < +∞⇔ −∞ < < +∞

Per effettuare la marginalizzazione riscriviamo come segue la ( ), ,W Zf w z :

( ) ( )

( ) ( ) ( )( )

( )( )

( )

2 2

2 2

2 22 22 2

2 2 2 2 2 2 2 2

2

2 2 2 2

2

, ,

12

1exp2

1 1exp22

W W Z

X Y

X YX Y

X Y X YY X X Y

X Y X Y X Y X Y

X Y

X Y X Y

X

f w f w z dz

w ww z zdz

w

σ σπσ σσ σ

μ μ μ μσ μ σ μσ σ σ σ σ σ σ σ

μ μ

π σ σ σ σ

σ

+∞

−∞

+∞

−∞

= ⋅ =

+= ⋅

+

⎧ ⎫⎡ ⎤− + − +⎡ ⎤ ⎡ ⎤− − −⎪ ⎪⎣ ⎦ ⎣ ⎦⎢ ⎥− + − + =⎨ ⎬⎢ ⎥+ +⎪ ⎪⎣ ⎦⎩ ⎭⎧ ⎫⎡ ⎤− +⎪ ⎪⎢ ⎥= − ⋅⎨ ⎬

⎢ ⎥+ +⎪ ⎪⎣ ⎦⎩ ⎭

∫

∫

( ) ( ) ( )

( )

( ) ( )

22 22 22

2 2 2 2 2 22 2

2

2 2 2 2

2 22 22 2

2 22 2

1 1exp22

1 1exp22

1 1exp22

X YY X X YY

X Y X Y X YX Y

X Y

X Y X Y

Y X X YX Y

X Y XX Y

ww z zdz

w

z w z

μ μσ μ σ μσπ σ σ σ σ σ σσ σ

μ μ

π σ σ σ σ

σ μ σ μσ σπ σ σ σσ σ

+∞

−∞

⎧ ⎫⎡ ⎤− +⎡ ⎤− − −+ ⎪ ⎪⎣ ⎦⎢ ⎥− + − =⎨ ⎬⎢ ⎥+⎪ ⎪⎣ ⎦⎩ ⎭⎧ ⎫⎡ ⎤− +⎪ ⎪⎢ ⎥= − ⋅⎨ ⎬

⎢ ⎥+ +⎪ ⎪⎣ ⎦⎩ ⎭

− −⎡ ⎤ −+ ⎣ ⎦− +

∫

( )( )

( )

( ) ( )

22 2

2 2 2 2 2 2

22 2

2 2 2 2 2 2

2 22 2 2 2 2 2 2 2 2

2 2 2 2

1 1 1exp2 22

2 21exp2

X Y X Y

Y X Y X Y

X Y X Y

X Y X Y X Y

XY Y X Y X X X Y X Y

X Y X Y

w

w

z w z w z z

σ σ μ μ

σ σ σ σ σ

μ μ σ σππ σ σ σ σ σ σ

σσ σ μ σ μ σ σ μ σ μσ σ σ σ

+∞

−∞

+∞

−∞

⎧ ⎫⎡ ⎤− +⎡ ⎤⎪ ⎪⎣ ⎦⎢ ⎥− =⎨ ⎬⎢ ⎥+⎪ ⎪⎣ ⎦⎩ ⎭⎧ ⎫⎡ ⎤− + +⎪ ⎪⎢ ⎥= − ⋅ ⋅⎨ ⎬

⎢ ⎥+ +⎪ ⎪⎣ ⎦⎩ ⎭

+ − − − + −− + −

∫

∫

( )( )

22

2 2 2 2Y X Y

X Y X Y

wdz

σ μ μ

σ σ σ σ

⎧ ⎫⎡ ⎤− +⎡ ⎤⎪ ⎪⎣ ⎦⎢ ⎥ =⎨ ⎬⎢ ⎥+⎪ ⎪⎣ ⎦⎩ ⎭

:

( )( )

( )

( )

( )

2 2 2 2 2 2 2

22 2 2

222

2

2 2 22 2 2 2 2

1 1 2 2 22 2 2

2112 1 22 2

12 2

1 1 1, ,2 2 2

2 12 2 2

1 12 2 2 1 2

12 2

w u u w wu u u w wu u

W U

ww w wu u

u wu ww

w

f w u e e e

e e

e e

e

π π π

π π

π π

π

⎡ ⎤− ⎡ ⎤ ⎡ ⎤− + − +⎢ ⎥− + − + −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ ⎤− ⎢ ⎥ ⎡ ⎤− − − +⎢ ⎥ ⎣ ⎦⎣ ⎦

⎡ ⎤− +⎡ ⎤ ⎢ ⎥−− ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡− ⎢

⎣

= = = =

= ⋅ =⋅

= ⋅ =⋅ ⋅

=⋅ ( )

( )2212 1 21

2 1 2

u w

eπ

⎡ ⎤−⎤ ⎢ ⎥−⎥ ⎢ ⎥⎢ ⎥⎦ ⎣ ⎦⋅⋅

Quindi marginalizziamo:

( ) ( )

( )

( )

( )

22

22

2

2112 1 22 2

2112 1 22 2

12 2

1 12 2 2 1 2

1 12 2 2 1 2

12 2

u ww

W

u ww

w

f w e e du

e e du

e

π π

π π

π

⎡ ⎤−⎡ ⎤ ⎢ ⎥−+∞ − ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

−∞

⎡ ⎤−⎡ ⎤ ⎢ ⎥−+∞− ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

−∞

⎡ ⎤− ⎢ ⎥

⎢ ⎥⎣ ⎦

= ⋅ ⋅ =⋅ ⋅

= ⋅ ⋅ =⋅

=⋅

∫

∫

Risulta quindi ( )0, 2W N∼ [W=X+Y ha ovviamente supporto ( ),−∞ +∞ ] Nota:

( )

( )2212 1 21 1

2 1 2

u w

e duπ

⎡ ⎤−⎢ ⎥−+∞⎢ ⎥⎣ ⎦

−∞

⋅ ⋅ =∫ .

La funzione integranda è la pdf di una Normale di media 2w e varianza 1

2 1,

2 2wU N⎡ ⎤⎛ ⎞

⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦∼ .

Somma di 2 variabili Esponenziali (s-indipendenti) di uguale parametro λ Z X Y= +

( )( )

; 0

; 0

xX

yY

f x e x

f y e y

−

−

= ≤ < +∞

= ≤ < +∞

λ

λ

λ

λ

( ) ( ) ( )( )

0

2

0

2

0

2

0

2

1

z

Z Y

z z x x

z z

zz

z

f z f z x f x dx

e e dx

e dx

e dx

z e

− − −

−

−

−

= − ⋅ ⋅ =

= ⋅ ⋅ ⋅ =

= ⋅ ⋅ =

= ⋅ ⋅ =

= ⋅ ⋅

∫∫∫∫

λ λ

λ

λ

λ

λ

λ

λ

λ

La Z è una variabile aleatoria gamma di parametro di scala λ e parametro di forma 2, risulta infatti:

( ) ( )( )

2 12

2

zz

Z

z ef z z e

− −− ⋅ ⋅ ⋅

= ⋅ ⋅ =Γ

λλ λ λ

λ

Somma di 2 variabili Esponenziali (s-indipendenti) di parametri xλ e yλ con

x yλ λ≠ . Z X Y= +

( )( )

; 0

; 0

x

y

xX x

yY y

f x e x

f y e y

λ

λ

λ

λ

−

−

= ≤ < +∞

= ≤ < +∞

( ) ( ) ( )( )

( )

0

0

0

y x

x yy

z

Z Y

z z x xx y

z xzx y

f z f z x f x dx

e e dx

e e dx

λ λ

λ λλ

λ λ

λ λ

− − −

− − ⋅−

= − ⋅ ⋅ =

= ⋅ ⋅ ⋅ ⋅ =

= ⋅ ⋅ ⋅

∫∫

∫

la quale ultima espressione se x yλ λ≠ risulta uguale a

( ) ( )0

1x y x yy y

y x

zx zz zx y x y

x y x y

zx y z

x y

e e e e

e e

λ λ λ λλ λ

λ λ

λ λ λ λλ λ λ λ

λ λλ λ

− − ⋅ − − ⋅− −

− −

⋅ ⋅⎡ ⎤ ⎡ ⎤⋅ − = ⋅ − =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦− −

⋅ ⎡ ⎤= ⋅ −⎣ ⎦−

Si vede facilmente che risulta: ( ) 0 0Zf z z≥ ∀ ≥ .

Si può inoltre verificare che ( )0

1Zf z dz+∞

⋅ =∫ .

Verifica

( )0 0

00

1 1

y x

y x

z zx y zZ

x y

zx y z

x y y x

f z dz e e dz

e e

λ λ

λ λ

λ λλ λ

λ λλ λ λ λ

+∞ − −

+∞ +∞− −

⋅⋅ = ⋅ − ⋅ =

−

⎧ ⎫⎡ ⎤⋅ ⎡ ⎤⎪ ⎪= − − − =⎢ ⎥⎨ ⎬⎢ ⎥− ⎢ ⎥ ⎣ ⎦⎪ ⎪⎣ ⎦⎩ ⎭

∫ ∫

1 1 1 10 0

1

x y x y

x y y x x y y x

x y x y

x y x y

λ λ λ λλ λ λ λ λ λ λ λ

λ λ λ λλ λ λ λ

⎧ ⎫⎡ ⎤ ⎧ ⎫⋅ ⋅⎡ ⎤⎪ ⎪ ⎪ ⎪= − + − − + = − =⎢ ⎥⎨ ⎬ ⎨ ⎬⎢ ⎥− −⎢ ⎥ ⎪ ⎪⎣ ⎦⎪ ⎪⎣ ⎦ ⎩ ⎭⎩ ⎭⋅ −

= ⋅ =− ⋅

Somma di 2 variabili Uniformi (0,1) s-indipendenti. Z X Y= +

( )( )

1; 0 1

1; 0 1X

Y

f x x

f y y

= ≤ ≤

= ≤ ≤

( ), , 1; 0 1, 0 1X Yf x y x y= ≤ ≤ ≤ ≤

per calcolare l’fdp di Z opero cambiamento di variabili da X ed Y a Z e U e poi marginalizzo: Z X Y X Z U

U Y Y U= + = −

⇔= =

1 11 1

0 1

X XZ UJY YZ U

∂ ∂−∂ ∂= = = =

∂ ∂∂ ∂

( ) ( ) ( )( )( ) ( )

, ,, , , , 1;

0 1, 0 1 0 2, 0, 1 1,per 0 1 0per 1 2 1 1

Z U X Yf z u f x z u y z u J

z u u z Max z u Min zz u zz z u

= ⋅ =

≤ − ≤ ≤ ≤ ⇔ ≤ ≤ − ≤ ≤

≤ ≤ ≤ ≤⇔

≤ ≤ − ≤ ≤

la marginalizzazione deve pertanto essere fatta trattando separatamente i casi 0 1z≤ ≤ e 1 2z< ≤ :

per 0 1z≤ ≤ si ottiene: ( ),0 0, 1

z z

Z Uf z u du du z⋅ = ⋅ =∫ ∫

mentre per 1 2z< ≤ si ottiene ( ) ( )

1 1

,1 1, 1 1 1 2Z Uz z

f z u du du z z− −

⋅ = ⋅ = − − = −∫ ∫

ne risulta:

( )z per 0 12-z per 1 2Z

zf z

z≤ ≤⎧

= ⎨ < ≤⎩

Verifica:

( )

1 22 21 2

0 10 1

2 22 2

1 1 1 30 4 2 2 2 12 2 2 2

z zz dz z dz z⎡ ⎤ ⎡ ⎤

⋅ + − ⋅ = + − =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

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

∫ ∫

Nota Z ha una distribuzione detta triangolare

Somma di Normali standard s-dipendenti con congiunta Normale W X Y= +

( ) ( )( )2 22

1 22 1

2

1, , ; , .2 1

x xy y

X Yf x y e x yρ

ρ

π ρ

− − +−

= −∞ < < ∞ −∞ < < ∞−

( ) ( )

( ) ( )

2

2

2,

2,

1, ; .2

1, ; .2

x

X X Y

y

Y X Y

f x f x y dy e x

f y f x y dx e y

π

π

−+∞

−∞

−+∞

−∞

= ⋅ = −∞ < < ∞

= ⋅ = −∞ < < ∞

∫

∫

Per calcolare la fdp di W opero cambiamento di variabili da X ,Y a W,U e poi marginalizzo:

W X Y X W UU Y Y U= + = −

⇔= =

⇒ 1 1

1 10 1

X XW UJY YW U

∂ ∂−∂ ∂= = = =

∂ ∂∂ ∂

( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( )

2 22

2 22

1 22 1

, , 2

1 2 1 2 12 1

2

1, , , , 12 1

1

2 1

w u w u u u

W U X Y

w wu u

f w u f x w u y w u J e

e

ρρ

ρ ρρ

π ρ

π ρ

⎡ ⎤− − − − +⎢ ⎥⎣ ⎦−

⎡ ⎤− − + + +⎣ ⎦−

= ⋅ = ⋅ =−

=−

( ) ( ) ( ) ( ) ( )2 22

1 2 1 2 12 1

2

1, ,2 1

w wu u

W W Uf w f w u du e duρ ρ

ρ

π ρ

⎡ ⎤− − + + +⎣ ⎦−+∞ +∞

−∞ −∞= ⋅ = ⋅

−∫ ∫

La variabile U ha supporto, ( ),−∞ +∞ , che non dipende da W. , , ,w u u w w u w u−∞ < − < +∞ −∞ < < +∞ −∞ < < +∞ −∞ < < +∞⇔ −∞ < < +∞

Per effettuare la marginalizzazione riscriviamo come segue la ( ), ,W Uf w u :

( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( )

( )( )

( )

2 22

22 2 22

22 222 2

1 2 1 2 12 1

1 11 2 1 2 12 2 12 12 2 1

2 1 2 1112 2 11 1 12 2 1

2 1, ,2 2 2 1 1

1 112 2 1 2

2

1 112 2 1 2

2

w wu u

W U

ww w wu u

wu uw ww

f w u e

e e

e e

ρ ρρ

ρ ρρρρ

ρ ρρρ ρρ

π π ρ ρ

ρπ ρ π

ρπ ρ π

⎡ ⎤− − + + +⎣ ⎦−

⎡ ⎤⎡ ⎤ ⎡ ⎤− − + + + + ⎢ ⎥− ⎢ ⎥ ⎣ ⎦ +− ⎢ ⎥+ ⎣ ⎦⎢ ⎥⎣ ⎦

+ +⎡ ⎤ − − − +− ⎢ ⎥ +− − −+⎢ ⎥⎣ ⎦

= =+ −

= =−+

= ⋅−+

( )2ρ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ =

( )( )

( )( ) ( ) ( )

( )( )

( )( )

( )( )

( )( )

2 222

2 22

22

11 2 212 1 12 12 2 1

2112 1 22 2 1

2112 1 22 2 1

1 112 2 1 2

2

1 112 2 1 2

2

1 112 2 1 2

2

w wu uw

w wu uw

u ww

e e

e e

e e

ρρ ρρρ

ρρ

ρρ

ρπ ρ π

ρπ ρ π

ρπ ρ π

⎡ ⎤+⎡ ⎤ ⎢ ⎥− − +− ⎢ ⎥ ⎢ ⎥− −−+⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤− +⎡ ⎤ ⎢ ⎥−− ⎢ ⎥ −⎢ ⎥+⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤−⎡ ⎤⎡ ⎤ ⎣ ⎦⎢ ⎥−− ⎢ ⎥ ⎢ ⎥−+⎢ ⎥⎣ ⎦ ⎣ ⎦

= ⋅ =−+

⋅ =−+

= ⋅−+

Quindi marginalizziamo:

( )( )

( )( )

( )

( )( )

( )( )

( )( )

22

22

2

2112 1 22 2 1

2112 1 22 2 1

12 2 1

1 112 2 1 2

2

1 112 2 1 2

2

12 2 1

u ww

W

u ww

w

f w e e du

e e du

e

ρρ

ρρ

ρ

ρπ ρ π

ρπ ρ π

π ρ

⎡ ⎤−⎡ ⎤⎡ ⎤ ⎣ ⎦⎢ ⎥−+∞ − ⎢ ⎥ ⎢ ⎥−+⎢ ⎥⎣ ⎦ ⎣ ⎦

−∞

⎡ ⎤−⎡ ⎤⎡ ⎤ ⎣ ⎦⎢ ⎥−+∞− ⎢ ⎥ ⎢ ⎥−+⎢ ⎥⎣ ⎦ ⎣ ⎦

−∞

⎡ ⎤− ⎢ ⎥

+⎢ ⎥⎣ ⎦

= ⋅ ⋅ =−+

= ⋅ ⋅ =−+

=+

∫

∫

Risulta quindi ( )0,2 1W N ρ+⎡ ⎤⎣ ⎦∼ [W=X+Y ha ovviamente supporto ( ),−∞ +∞ ] Nota 1: A conferma del risultato ottenuto consideriamo che: ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

0

2 ,

E W E X Y E X E Y

Var W Var X Y Var X Var Y Cov X Y

= + = + =

= + = + +

Da cui essendo:

( ) ( ) ( )( ) ( )

( ) ( ), ,1 e ,

1Cov X Y Cov X Y

Var X Var Y Cov X YVar X Var Y

ρ= = = = =⋅

si ottiene: ( ) ( )1 1 2 2 1Var W ρ ρ= + + = + Nota 2

( )( )

2212 1 21 1

122

u w

e duρ

ρπ

⎡ ⎤−⎡ ⎤⎣ ⎦⎢ ⎥−+∞ ⎢ ⎥−⎣ ⎦

−∞

⋅ =−∫ .

La funzione integranda è la pdf di una Normale di media 2w e varianza 1

2ρ− 1,

2 2wU N ρ⎡ − ⎤⎛ ⎞

⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦∼ .