=a ∫ 0 1 x ⅆ x + ∫ 0 1 ⅆ x + ∫ 1 2 ⅆ x - ∫ 1 2 x ⅆ x +a ∫ 2 3 ⅆ x +4a ∫ 3 4 ⅆ x -a ∫ 3 4 x ⅆ x =a [ x 2 2 ] 0 1 + [x] 0 1 +( a+2 ) [x] 1 2 - [ x 2 2 ] 1 2 +a [x] 2 3 +4a [x] 3 4 -a [ x 2 2 ] 3 4 = = a 2 +1+( a+2 )-2+ 1 2 +a+4a-8a+ 9 2 a=a( 1 2 +1+1+4-8+ 9 2 )+( 1+ 1 2 )=3a+ 3 2 =1 ⇒ a= 1- 3 2 3 =- 1 6 Quindi la distribuzione di probabilità diventa (normalizzata): f(x)= { 0 x<0 - x 6 +1 0 ≤ x<1 -x+ 11 6 1 ≤ x<2 - 1 6 2 ≤ x<3 x 6 - 2 3 3 ≤ x<4 0 4 ≤ x
In questo caso: F(x) ={ 0 x<0 ∫ 0 x ( - x 6 +1 ) ⅆ x 0 ≤ x<1 F(1)+ ∫ 1 x ( -x+ 11 6 ) ⅆ x 1 ≤ x<2 F(2)+ ∫ 2 x ( - 1 6 ) ⅆ x 2 ≤ x<3 F(3)+ ∫ 3 x ( x 6 - 2 3 ) ⅆ x 3 ≤ x<4 F(4) 4 ≤ x ={ 0 x<0 - x 2 12 +x 0 ≤ x<1 F(1)+ [ - x 2 2 + 11 6 x ] 1 x 1 ≤ x<2 F(2)+ [ - x 6 ] 2 x 2 ≤ x<3 F(3)+ [ x 2 12 - 2 3 x ] 3 x 3 ≤ x<4 F(4) 4 ≤ x ={ 0 x<0 - x 2 12 +x 0 ≤ x<1 -1 12 +1+[ ( - x 2 2 + 11 6 x )-( - 1 2 + 11 6 ) ] 1 ≤ x<2 F(2)+[ ( - x 6 )-( - 2 6 ) ] 2 ≤ x<3 F(3)+[ ( x 2 12 - 2 3 x )-( 9 12 -2 ) ] 3 ≤ x<4 F(4) 4 ≤ x = ={ 0 x<0 - x 2 12 +x 0 ≤ x<1 -1 12 +1- x 2 2 + 11 6 x+ 1 2 - 11 6 1 ≤ x<2 F(2)- x 6 + 2 6 2 ≤ x<3 F(3)+ x 2 12 - 2 3 x- 9 12 +2 3 ≤ x<4 F(4) 4 ≤ x ={ 0 x<0 - x 2 12 +x 0 ≤ x<1 - x 2 2 + 11 6 x- 5 12 1 ≤ x<2 - 4 2 + 22 6 - 5 12 - x 6 + 2 6 2 ≤ x<3 F(3)+ x 2 12 - 2 3 x- 9 12 +2 3 ≤ x<4 F(4) 4 ≤ x ={ 0 x<0 - x 2 12 +x 0 ≤ x<1 - x 2 2 + 11 6 x- 5 12 1 ≤ x<2 - x 6 + 19 12 2 ≤ x<3 - 3 6 + 19 12 + x 2 12 - 2 3 x- 9 12 +2 3 ≤ x<4 F(4) 4 ≤ x ={ 0 x<0 - x 2 12 +x 0 ≤ x<1 - x 2 2 + 11 6 x- 5 12 1 ≤ x<2 - x 6 + 19 12 2 ≤ x<3 x 2 12 - 2 3 x+ 7 3 3 ≤ x<4 1 4 ≤ x Di seguito il grafico della funzione di ripartizione.
In questo caso: F(x) ={ 0 x<0 ∫ 0 x ( - x 6 +1 ) ⅆ x 0 ≤ x<1 F(1)+ ∫ 1 x ( -x+ 11 6 ) ⅆ x 1 ≤ x<2 F(2)+ ∫ 2 x ( - 1 6 ) ⅆ x 2 ≤ x<3 F(3)+ ∫ 3 x ( x 6 - 2 3 ) ⅆ x 3 ≤ x<4 F(4) 4 ≤ x ={ 0 x<0 - x 2 12 +x 0 ≤ x<1 F(1)+ [ - x 2 2 + 11 6 x ] 1 x 1 ≤ x<2 F(2)+ [ - x 6 ] 2 x 2 ≤ x<3 F(3)+ [ x 2 12 - 2 3 x ] 3 x 3 ≤ x<4 F(4) 4 ≤ x ={ 0 x<0 - x 2 12 +x 0 ≤ x<1 -1 12 +1+[ ( - x 2 2 + 11 6 x )-( - 1 2 + 11 6 ) ] 1 ≤ x<2 F(2)+[ ( - x 6 )-( - 2 6 ) ] 2 ≤ x<3 F(3)+[ ( x 2 12 - 2 3 x )-( 9 12 -2 ) ] 3 ≤ x<4 F(4) 4 ≤ x =
={ 0 x<0 - x 2 12 +x 0 ≤ x<1 -1 12 +1- x 2 2 + 11 6 x+ 1 2 - 11 6 1 ≤ x<2 F(2)- x 6 + 2 6 2 ≤ x<3 F(3)+ x 2 12 - 2 3 x- 9 12 +2 3 ≤ x<4 F(4) 4 ≤ x ={ 0 x<0 - x 2 12 +x 0 ≤ x<1 - x 2 2 + 11 6 x- 5 12 1 ≤ x<2 - 4 2 + 22 6 - 5 12 - x 6 + 2 6 2 ≤ x<3 F(3)+ x 2 12 - 2 3 x- 9 12 +2 3 ≤ x<4 F(4) 4 ≤ x ={ 0 x<0 - x 2 12 +x 0 ≤ x<1 - x 2 2 + 11 6 x- 5 12 1 ≤ x<2 - x 6 + 19 12 2 ≤ x<3 - 3 6 + 19 12 + x 2 12 - 2 3 x- 9 12 +2 3 ≤ x<4 F(4) 4 ≤ x ={ 0 x<0 - x 2 12 +x 0 ≤ x<1 - x 2 2 + 11 6 x- 5 12 1 ≤ x<2 - x 6 + 19 12 2 ≤ x<3 x 2 12 - 2 3 x+ 7 3 3 ≤ x<4 1 4 ≤ x
m= ∫ - ∞ + ∞ x ⋅ f(x) ⅆ x = ∫ 0 1 ( - x 6 +1 )x ⅆ x + ∫ 1 2 ( -x+ 11 6 )x ⅆ x + ∫ 2 3 ( - 1 6 )x ⅆ x + ∫ 3 4 ( x 6 - 2 3 )x ⅆ x =
=- 1 6 ∫ 0 1 x 2 ⅆ x + ∫ 0 1 x ⅆ x - ∫ 1 2 x 2 ⅆ x + 11 6 ∫ 1 2 x ⅆ x - 1 6 ∫ 2 3 x ⅆ x + 1 6 ∫ 3 4 x 2 ⅆ x - 2 3 ∫ 3 4 x ⅆ x =- 1 6 [ x 3 3 ] 0 1 + [ x 2 2 ] 0 1 - [ x 3 3 ] 1 2 + 11 6 [ x 2 2 ] 1 2 - 1 6 [ x 2 2 ] 2 3 + 1 6 [ x 3 3 ] 3 4 - 2 3 [ x 2 2 ] 3 4 =
=- 1 18 + 1 2 -[ 8 3 - 1 3 ]+ 11 6 [ 4 2 - 1 2 ]- 1 6 [ 9 2 - 4 2 ]+ 1 6 [ 64 3 - 27 3 ]- 2 3 [ 16 2 - 9 2 ]=- 1 18 + 1 2 - 8 3 + 1 3 + 22 6 - 11 12 - 3 4 + 1 3 + 32 9 - 3 2 - 16 3 +3= -2+18-96+12+132-33-27+12+128-54-192+108 36 = 6 36 = 1 6
σ 2 = ∫ - ∞ + ∞ ( x-m ) 2 ⋅ f(x) ⅆ x = ∫ 0 1 ( - x 6 +1 ) ( x- 1 6 ) 2 ⅆ x + ∫ 1 2 ( -x+ 11 6 ) ( x- 1 6 ) 2 ⅆ x + ∫ 2 3 ( - 1 6 ) ( x- 1 6 ) 2 ⅆ x + ∫ 3 4 ( x 6 - 2 3 ) ( x- 1 6 ) 2 ⅆ x =
= ∫ 0 1 ( - x 6 +1 )( x 2 + 1 36 - x 3 ) ⅆ x + ∫ 1 2 ( -x+ 11 6 )( x 2 + 1 36 - x 3 ) ⅆ x + ∫ 2 3 ( - 1 6 )( x 2 + 1 36 - x 3 ) ⅆ x + ∫ 3 4 ( x 6 - 2 3 )( x 2 + 1 36 - x 3 ) ⅆ x =
= ∫ 0 1 ( - x 3 6 - x 216 + x 2 18 + x 2 + 1 36 - x 3 ) ⅆ x + ∫ 1 2 ( -x 3 - x 36 + x 2 3 + 11 6 x 2 + 11 216 - 11 18 x ) ⅆ x - 1 6 ∫ 2 3 ( x 2 + 1 36 - x 3 ) ⅆ x + ∫ 3 4 ( x 3 6 + x 216 - x 2 18 - 2 3 x 2 - 1 54 + 2 9 x ) ⅆ x =
= ∫ 0 1 ( - x 3 6 + 19 18 x 2 - 73 216 x+ 1 36 ) ⅆ x + ∫ 1 2 ( - x 3 + 13 6 x 2 - 23 36 x+ 11 216 ) ⅆ x - 1 6 ∫ 2 3 ( x 2 + 1 36 - x 3 ) ⅆ x + ∫ 3 4 ( x 3 6 - 13 18 x 2 + 49 216 x- 1 54 ) ⅆ x =
= [ - x 4 24 + 19 54 x 3 - 73 432 x 2 + 1 36 x ] 0 1 + [ - x 4 4 + 13 18 x 3 - 23 72 x 2 + 11 216 x ] 1 2 - 1 6 [ x 3 3 + x 36 - x 2 6 ] 2 3 + [ x 4 24 - 13 54 x 3 + 49 432 x 2 - x 54 ] 3 4 =
=- 1 24 + 19 54 - 73 432 + 1 36 +( - 16 4 + 104 18 - 92 72 + 22 216 )-( - 1 4 + 13 18 - 23 72 + 11 216 )- 1 6 [ ( 27 3 + 3 36 - 9 6 )-( 8 3 + 2 36 - 4 6 ) ]+( 256 24 - 832 54 + 784 432 - 4 54 )-( 81 24 - 351 54 + 441 432 - 3 54 )=
=- 1 24 + 19 54 - 73 432 + 1 36 - 16 4 + 104 18 - 92 72 + 22 216 + 1 4 - 13 18 + 23 72 - 11 216 - 27 18 - 3 216 + 9 36 + 8 18 + 2 216 - 4 36 + 256 24 - 832 54 + 784 432 - 4 54 - 81 24 + 351 54 - 441 432 + 3 54 =
= -18+152-73+12-1728+2496-552+44+108-312+138-22-648-6+108+192+4-48+4608-6656+784-32-1458+2808-441+24 432 =- 516 432 =- 43 36
p( x ≤ 2 )=F(2)=- 2 6 + 19 12 = -4+19 12 = 15 12