E s e rc it a z io n e 4 - S o lu z io n e 1 . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) [ ] 37, 2 64, 2 04, 0 6 15, 0 1 10, 0 0
37, 2 04, 0 64, 2 6 15, 0 64, 2 1 10, 0 64, 2 0
64, 2 04, 0 6 15, 0 1 10, 0 0 45, 0 20, 0 15, 0 10, 0 2 71, 0 06, 0 20, 0 25, 0 20, 0 5 2 55, 0 45, 0 1 3 1 3 45, 0 20, 0 15, 0 10, 0 3 7, 0 25, 0 20, 0 15, 0 10, 0 3
22222222 2222= − ⋅ + + ⋅ + ⋅ = − = − =
= ⋅ − + + ⋅ − + ⋅ − = − =
= ⋅ + + ⋅ + ⋅ = = = + + = ≤ = + + + = ≤ ≤ = − = < − = ≥ = + + = < = + + + = ≤ ∑ ∑ ∑ ∑ K
K
K x xf x f x X E X E X V
x f x E x X V
x xf X E
X P
X P
X P X P
X P
X P
2 . L a v .c . h a d is tr ib u z io n e b in o m ia le c o n n = 1 0 e π = 0 ,0 5 . D a lla s i o tt ie n e l a s e g u e n te t a b e lla : O p p u re , ri c o rd a n d o i l v a lo re a tt e s o e l a v a ri a n z a d i u n a b in o m ia le :
( ) ( ) ( )
xxx x f x X P
−−
= = =
1005, 0 1 05, 0 10
0,00000,00000,00000,00000,00000,00010,00100,01050,07460,31510,5987f(x)109876543210X
( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) 475, 0 0000, 0 5, 0 10 3151, 0 5, 0 1 5987, 0 5, 0 0
5, 0 0000, 0 10 3151, 0 1 5987, 0 0
9138, 0 1
22221 0
= ⋅ − + + ⋅ − + ⋅ − = − =
= ⋅ + + ⋅ + ⋅ = =
= = = ≤ ∑ ∑ ∑
=K
K x f x E x X V
x xf X E
x X P X P
x( ) ( ) ( ) 475, 0 1 5, 0 = − ⋅ ⋅ = = ⋅ = π π π n X V
n X E 3 . L a v .c . X h a d is tr ib u z io n e d i P o is s o n c o n λ = 8 , q u in d i la f u n z io n e d i p ro b a b ili tà è d a ta d a In o lt re E (X ) = V (X ) = 8 . Q u in d i: ( ) ( ) !
8
8x e x f x X P
x−
= = = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 8284, 2
8 5254, 0 1912, 0 1 10 1 10
5254, 0 ! 9
8 !8
8 ! 7
8 ! 6
8 9 6
1912, 0 !5
8 !1
8 ! 0
8 5
898887869 68581805 0
=
= + − = < − = ≥
= + + + = = = ≤ ≤
= + + + = = = ≤
−−−− =−−− =
∑
∑ X V X E
X P X P
e e e e x X P X P
e e e x X P X P
xx
K
4 . D a ta X ∼N (1 ,2 5 ;0 ,4 6
2), s i p u ò c a lc o la re l a v .c . n o rm a le s ta n d a rd iz z a ta . Q u in d i, c o n l ’u ti liz z o d e lle t a v o le p o s s ia m o c a lc o la re l e s e g u e n ti p ro b a b ili tà : In b a s e a i ri s u lt a ti d e i p u n ti p re c e d e n ti è im m e d ia to o tt e n e re P (X ≤2 )= 0 ,9 4 8 5 . D i c o n s e g u e n z a l a p ro b a b ili tà c h e u n i n d iv id u o a b b ia u n i n c id e n te è p a ri a p = 1 -0 ,9 4 8 5 = 0 ,0 5 1 5 . In d ic a n d o c o n Y l a v .c . c h e e s p ri m e i l n u m e ro d i a s s ic u ra ti c h e h a n n o u n in c id e n te s u 1 0 0 a s s ic u ra ti , c io è Y ∼ B in (1 0 0 ; 0 ,0 5 1 5 ), l a p ro b a b ili tà c h e p iù d i 5 in d iv id u i a b b ia n o u n i n c id e n te è p a ri a
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 6540, 0 2945, 0 9485, 0 54, 0 63, 1 46, 0 25, 1 1 46, 0 25, 1 2 2 1 2945, 0 54, 0 54, 0 46, 0 25, 1 1 1 = − = − Φ − Φ = = − ≤ − − ≤ = ≤ ≤ = Φ = − Φ = − < = ≤ Z P Z P X P
Z P X P ( ) ( ) ( ) 4110, 0 0515, 0 1 0515, 0 100 1 5 1 5
5 0
100
= −
− = ≤ − = > ∑
=− y
yy
y Y P Y P
5 . U ti liz z a n d o l e t a v o le s i ri c a v a c h e i q u a n ti li d e lla d is tr ib u z io n e n o rm a le s ta n d a rd iz z a ta c o rr is p o n d e n ti a l 2 5 -e s im o e a l 9 0 -e s im o p e rc e n ti le s o n o - 0 ,6 7 e 1 ,2 8 . Q u in d i i p a ra m e tr i µ e σ d e lla d is tr ib u z io n e n o rm a le s i o tt e n g o n o ri s o lv e n d o i n s e g u e n te s is te m a d i e q u a z io n i: d a c u i µ = 1 ,5 3 7 4 e σ = 0 ,2 0 5 1 . Il v a lo re a tt e s o d e l d ia m e tr o d i u n b u llo n e è µ = 1 ,5 3 7 4 , m e n tr e u n a m is u ra d e ll’ in a c c u ra te z z a d e l p ro c e s s o è d a ta d a lla d e v ia z io n e s ta n d a rd σ = 0 ,2 0 5 1 . In o lt re : Il n u m e ro d i b u llo n i d i la rg h e z z a s u p e ri o re a 2 m m i n u n l o tt o d i 1 0 b u llo n i p u ò e s s e re d e s c ri tt o d a u n a v .c . Y ∼B in (1 0 ;0 ,0 1 2 0 5 ). Q u in d i
= − − = − 28, 1 8, 1
67, 0 4, 1 σ µ σ µ ( ) ( ) ( ) ( ) ( ) 01205, 0 255, 2 1 2051, 0 5374, 1 2 1 2
05, 0 645, 1 1 645, 1 2051, 0 5374, 1 2, 1 2, 1 =≈ Φ − =
− Φ − = >
≈ Φ − = − Φ =
− Φ = ≤ X P
X P ( ) ( ) 9938, 0 01205, 0 1 01205, 0 10 1
1 0
100
= −
= ≤ ∑
=− y
yy
