ÿþR a c h u n e k p r a w d o p o d o b i e Ds t w a M A P 1 0 6 4
W y d z i a B E l e k t r o n i k i , r o k a k a d . 2 0 0 8 / 0 9 , s e m . l e t n i
W y k Ba d o w c a : d r h a b . A . J u r l e w i c z
P r z y k Ba d y d o l i s t y 9 : W a r t o [ o c z e k i w a n a , w a r i a n c j a , m e d i a n a , k w a r t y l e
r o z k Ba d u p r a w d o p o d o b i e Ds t w a . S t a n d a r y z a c j a r o z k Ba d u n o r m a l n e g o .
P r z y k Ba d y d o z a d a n i a 9 . 1 :
( a ) W y l i c z y - o i l e t o m o |l i w e - w a r t o [ o c z e k i w a n i w a r i a n c j o r a z w y z n a c z y m e d i a n i k w a r t y l e
n 1 2 3 4
d y s k r e t n e g o r o z k Ba d u z m i e n n e j l o s o w e j X p o d a n e g o w t a b e l i : x n 2 3 4 5
p n 0 , 1 0 , 3 0 , 4 0 , 2
" E X = 2 · 0 , 1 + 3 · 0 , 3 + 4 · 0 , 4 + 5 · 0 , 2 = 3 , 7
" D 2 X = 2 2 · 0 , 1 + 3 2 · 0 , 3 + 4 2 · 0 , 4 + 5 2 · 0 , 2 - ( E X ) 2 = 0 , 8 1
"
( D 2 X = 0 , 9 )
" x 0 , 2 5 = 3 , x 0 , 5 = x 0 , 7 5 = 4
1 . 5
F ( x )
1
1
0 . 8
0 . 7 5
0 . 5
0 . 5
0 . 4
0 . 2 5
0 . 1
x
0
2 4 5
3
x 0 . 2 5
x 0 . 5
x 0 . 7 5
- 0 . 5
0 1 2 3 4 5
1
( b ) W y l i c z y - o i l e t o m o |l i w e - w a r t o [ o c z e k i w a n i w a r i a n c j o r a z w y z n a c z y m e d i a n i k w a r t y l e
d y s k r e t n e g o r o z k Ba d u z m i e n n e j l o s o w e j X , z a d a n e g o c i g i e m { ( x n , p n ) , n = 1 , 2 , . . . } , g d z i e
2
x n = 2 n , p n = , n = 1 , 2 , . . . .
3 n
" " 1
2
3
" E X = x n p n = 2 n · = 4 · = 3 .
2
1
3 n
n = 1 n = 1
1 -
3
"
x
( S k o r z y s t a l i [m y z e w z o r u z A n a l i z y M a t e m a t y c z n e j : n x n = d l a | x | < 1 . )
( 1 - x ) 2
n = 1
" " " 1
2 8 1 n - 1 8 1 +
3
" D 2 X = x 2 p n - ( E X ) 2 = ( 2 n ) 2 · - 3 2 = n 2 · - 9 = ·
3 - 9 = 3 .
n
1
3 n 3 3 3
n = 1 n = 1 n = 1
1 -
3
"
1 + x
( S k o r z y s t a l i [m y z e w z o r u z A n a l i z y M a t e m a t y c z n e j : n 2 x n - 1 = d l a | x | < 1 . )
( 1 - x ) 3
n = 1
"
( D 2 X H" 1 , 7 2 3 0 )
" x 0 , 2 5 = x 0 , 5 = 2 , x 0 , 7 5 = 4
1 . 5
F ( x )
1
2 6 / 2 7
8 / 9
0 . 7 5
2 / 3
0 . 5
0 . 5
0 . 2 5
x
0
6
2 4
x 0 . 2 5
x 0 . 7 5
x 0 . 5
- 0 . 5
- 1 0 1 2 3 4 5 6 7
2
( c ) W y l i c z y - o i l e t o m o |l i w e - w a r t o [ o c z e k i w a n i w a r i a n c j o r a z w y z n a c z y m e d i a n i k w a r t y l e
n 1 2 3
d y s k r e t n e g o r o z k Ba d u z m i e n n e j l o s o w e j X p o d a n e g o w t a b e l i : - 1 5 1 0
x n
p n 1 1 1
2 3 6
1 1 1 1 7
" E X = - 1 · + 5 · + 1 0 · = H" 2 , 8 3 3 3
2 3 6 6
1 1 1 6 2 9
" D 2 X = ( - 1 ) 2 · + 5 2 · + 1 0 2 · - ( E X ) 2 = H" 1 7 , 4 7 2 2
2 3 6 3 6
"
( D 2 X H" 4 , 1 7 8 0 )
" x 0 , 2 5 = - 1 , x 0 , 5 - d o w o l n a l i c z b a z p r z e d z i a Bu [ - 1 , 5 ] , x 0 , 7 5 = 5
1 . 5
F ( x )
1
1
5 / 6
0 , 7 5
0 , 5
0 . 5
1 / 2
0 , 2 5
0
5
- 1
1 0
p r z e d z i a l
x 0 , 7 5
x 0 , 2 5 m e d i a n
- 0 . 5
0 2 4 6 8 1 0
3
P r z y k Ba d y d o z a d a n i a 9 . 2 :
( a ) W y l i c z y - o i l e t o m o |l i w e - w a r t o [ o c z e k i w a n i w a r i a n c j o r a z w y z n a c z y m e d i a n i k w a r t y l e
"
3
0 d l a x " [ 0 ,
/
"3 ] ,
c i g Be g o r o z k Ba d u z m i e n n e j l o s o w e j X o g s t o [c i f ( x ) =
3
x 2 d l a x " [ 0 , 3 ] .
"
3
"
"
3
3
" 3
x 4 3 3 3
" E X = x f ( x ) d x = x 3 d x = = H" 1 , 0 8 1 7
4 4
- " 0
0
"
3
" " "
"
3
3 3 3
" 3
x 5 3 3 9 9 9 3 9
" D 2 X = x 2 f ( x ) d x - ( E X ) 2 = x 4 d x - ( E X ) 2 = = - = H" 0 , 0 7 8 0
5 5 1 6 8 0
- " 0
0
"
( D 2 X H" 0 , 2 7 9 3 )
" D y s t r y b u a n t a r o z k Ba d u X t o
ñø
ôø
0 d l a x 0 ,
ôø
òø
x
"
3
x 3
F ( x ) = f ( t ) d t = d l a 0 < x 3 ,
3
ôø
"
- " ôø
óø 3
1 d l a x > 3 .
"
3
" F ( x ) = q Ô! x 3 = 3 q Ô! x = 3 q d l a 0 < q < 1
" " "
3 3 3
" Z a t e m x 0 , 2 5 = 0 , 7 5 H" 0 , 9 0 8 6 , x 0 , 5 = 1 , 5 H" 1 , 1 4 4 7 , x 0 , 7 5 = 2 , 2 5 H" 1 , 3 1 0 4
( b ) W y l i c z y - o i l e t o m o |l i w e - w a r t o [ o c z e k i w a n i w a r i a n c j o r a z w y z n a c z y m e d i a n i k w a r t y l e
ñø
òø 0 d l a x 1 ,
c i g Be g o r o z k Ba d u z m i e n n e j l o s o w e j X o g s t o [c i f ( x ) = 1
óø
d l a x > 1 .
3 x 4 / 3
" "
d x
" x f ( x ) d x = - r o z b i e |n a d o "
3 x 4 / 3
- " 1
Z a t e m E X n i e j e s t s k o Dc z o n a , a D 2 X n i e i s t n i e j e
" D y s t r y b u a n t a r o z k Ba d u X t o
x
0 d l a x 1 ,
F ( x ) = f ( t ) d t =
1 - x - 1 / 3 d l a x > 1 .
- "
" F ( x ) = q Ô! 1 - x - 1 / 3 = q Ô! x = ( 1 - q ) - 3 d l a 0 < q < 1
" Z a t e m x 0 , 2 5 = 0 , 7 5 - 3 H" 2 , 3 7 0 4 , x 0 , 5 = 0 , 5 - 3 = 8 , x 0 , 7 5 = 0 , 2 5 - 3 H" 6 4
4
( c ) W y l i c z y - o i l e t o m o |l i w e - w a r t o [ o c z e k i w a n i w a r i a n c j o r a z w y z n a c z y m e d i a n i k w a r t y l e
ñø
ôø
0 d l a x < - 1 ,
ôø
ôø
ôø
6
ôø
ôø - ( x 2 - 4 ) d l a - 1 x < 1 ,
òø
5 9
c i g Be g o r o z k Ba d u z m i e n n e j l o s o w e j X m a r o z k Ba d o g s t o [c i f ( x ) = 0 d l a 1 x < 2 ,
ôø
ôø
6
ôø
ôø - ( x - 5 ) d l a 2 x < 3 ,
ôø
5 9
ôø
óø
0 d l a 3 x
" 1 3
6
" E X = x f ( x ) d x = - x ( x 2 - 4 ) d x + x ( x - 5 ) d x =
5 9
- " 2
ëø ëø
3 öø- 1
x 3 x 2
6 3 7
íø íø øø
= - 0 + - 5 = H" 0 , 6 2 7 1
5 9 5 9
3 2
2
( p i e r w s z a c a Bk a w s u m i e r ó w n a j e s t 0 j a k o c a Bk a z f u n k c j i n i e p a r z y s t e j p o p r z e d z i a l e s y -
m e t r y c z n y m w z g l d e m z e r a )
2
" 1 3
6 3 7
" D 2 X = x 2 f ( x ) d x - ( E X ) 2 = - x 2 ( x 2 - 4 ) d x + x 2 ( x - 5 ) d x - =
5 9 5 9
- " - 1 2
ëø ëø
1 öø ëø 3 öø
2
x 5 x 3 x 4 x 3
6 3 7 3 7 4 8 9 0 9
íø íø øø íø øø
= - 2 - 4 + - 5 - = H" 1 , 4 0 5 0
5 9 5 9 1 0 · 5 9 2
5 3 4 3
0 2
( w y k o r z y s t a l i [m y f a k t , |e p i e r w s z a c a Bk a j e s t z f u n k c j i p a r z y s t e j p o p r z e d z i a l e s y m e t r y c z -
n y m w z g l d e m z e r a )
"
( D 2 X H" 1 , 1 8 5 0 ) 1 . 2
F ( x )
" D y s t r y b u a n t a r o z k Ba d u X t o ( z p r z y k Ba d u 8 . 2 ) 1
ñø
ôø
0 d l a x < - 1 ,
ôø
0 . 8
ôø
ôø
0 , 7 5
ôø
H" 0 , 7 4 5 8
ôø 2 ( x ( 1 2 - x 2 ) + 1 1 )
ôø
ôø d l a - 1 x < 1 ,
ôø 0 . 6
5 9
òø
0 , 5
4 4
F ( x ) =
d l a 1 x < 2 ,
0 . 4
5 9
ôø
ôø
ôø
3 x ( 1 0 - x ) - 4
ôø
ôø
0 , 2 5
d l a 2 x < 3 ,
ôø
0 . 2
ôø 5 9
ôø
ôø
óø
- 1 3
1 d l a 3 x 2
1
0
H" 2 , 0 1 4
x 0 , 2 5 x 0 , 5 x 0 , 7 5 x
- 0 . 2
- 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5
2 ( x ( 1 2 - x 2 ) + 1 1 )
" F ( x ) = 0 , 2 5 Ô! = 0 , 2 5 ; - 1 < x < 1 Ô!
5 9
Ô! - x 3 + 1 2 x + 3 , 6 2 5 = 0 ; - 1 < x < 1
x 0 , 2 5 j e s t r o z w i z a n i e m t e g o r ó w n a n i a
M e t o d p r z y b l i |o n o t r z y m u j e m y r o z w i z a n i e x 0 , 2 5 H" - 0 , 3 1 2 5
2 ( x ( 1 2 - x 2 ) + 1 1 )
" F ( x ) = 0 , 5 Ô! = 0 , 5 ; - 1 < x < 1 Ô!
5 9
Ô! - x 3 + 1 2 x - 3 , 7 5 = 0 ; - 1 < x < 1
x 0 , 5 j e s t r o z w i z a n i e m t e g o r ó w n a n i a
M e t o d p r z y b l i |o n o t r z y m u j e m y r o z w i z a n i e x 0 , 5 H" 0 , 3 1 2 5
3 x ( 1 0 - x ) - 4
1 9 3
" F ( x ) = 0 , 7 5 Ô! = 0 , 7 5 ; 2 < x < 3 Ô! x 2 - 1 0 x + = 0 ; 2 < x < 3
5 9 1 2
"
1 0 7
1 0 -
1 0 7
3
" = , x 0 , 7 5 = H" 2 , 0 1 3 9
3 2
5
( d ) W y l i c z y - o i l e t o m o |l i w e - w a r t o [ o c z e k i w a n i w a r i a n c j o r a z w y z n a c z y m e d i a n i k w a r t y l e
ñø
ôø 0 , 2 5 e x d l a x 0 ,
òø
0 , 2 5
c i g Be g o r o z k Ba d u z m i e n n e j l o s o w e j X o g s t o [c i f ( x ) = d l a 0 < x l n 2 ,
l n 2
ôø
óø
e - x d l a x > l n 2
" 0 l n 2 "
0 , 2 5
" E X = x f ( x ) d x = 0 , 2 5 x e x d x + x d x + x e - x d x =
l n 2
- " - " 0 l n 2
ëø ëø
0 l n "
x 2 2
0 , 2 5 0 , 2 5 l n 2 l n 2 + 1
íø íø
= 0 , 2 5 ( x - 1 ) e x + + - ( x + 1 ) e - x = - 0 , 2 5 + + =
l n 2 2 2
2
- " 0 l n 2
= 0 , 6 2 5 l n 2 + 0 , 2 5 H" 0 , 6 8 3 2
O b l i c z e n i a p o m o c n i c z e :
x e x d x = x ( e x ) d x = x e x - e x d x = ( x - 1 ) e x + C
x e - x d x = x ( - e - x ) d x = - x e - x + e - x d x = - ( x + 1 ) e - x + C
H
x - 1 1
l i m ( x - 1 ) e x = l i m = l i m = 0
e - x - e - x
x ’!- " x ’!- " x ’!- "
H
x + 1 1
l i m ( x + 1 ) e - x = l i m = l i m = 0
e x e x
x ’! " x ’! " x ’! "
" 0 l n 2 "
0 , 2 5
" D 2 X = x 2 f ( x ) d x - ( E X ) 2 = 0 , 2 5 x 2 e x d x + x 2 d x + x 2 e - x d x =
l n 2
- " - " 0 l n
ëø ëø
0 l n 2
"
x 3 2
0 , 2 5 0 , 2 5 l n 2 2
íø íø
= 0 , 2 5 ( x 2 - 2 x + 2 ) e x + + - ( x 2 + 2 x + 2 ) e - x = - 0 , 5 + +
l n 2 3
3
- " 0 l n 2
2
3 . 5 l n 2 2
+ l n 2 + 2 l n 2 + 2 - ( 0 , 6 2 5 l n 2 + 0 , 2 5 ) 2 = 1 , 2 5 + 0 , 3 7 5 l n 2 + H" 1 , 7 9 0 2
2 6
O b l i c z e n i a p o m o c n i c z e :
x 2 e x d x = x 2 ( e x ) d x = x 2 e x - 2 x e x d x = ( x 2 - 2 x + 2 ) e x + C
x 2 e - x d x = x 2 ( - e - x ) d x = - x 2 e - x + 2 x e - x d x = - ( x 2 + 2 x + 2 ) e - x + C
H
x 2 - 2 x + 2 2 x - 2
l i m ( x 2 - 2 x + 2 ) e x = l i m = l i m = 0
e - x - e - x
x ’!- " x ’!- " x ’!- "
H
x 2 + 2 x + 2 2 x + 2
l i m ( x 2 + 2 x + 2 ) e - x = l i m = l i m = 0
e x e x
x ’! " x ’! " x ’! "
"
( D 2 X H" 1 , 3 3 8 0 )
" D y s t r y b u a n t a r o z k Ba d u X t o ( z p r z y k Ba d u 8 . 3 ( c ) )
1 . 5
F ( x )
ñø
ôø
0 , 2 5 e x d l a x 0 ,
ôø
òø
1
x
F ( x ) = 0 , 2 5 + 1 d l a 0 < x l n 2 ,
l n 2
ôø
ôø
óø 0 , 7 5
1 - e - x d l a x > l n 2
0 . 5
0 , 5
0 , 2 5
x
0
" F ( x ) = 0 , 2 5 Ô! x = 0
= l n 4
x 0 , 7 5
x 0 , 2 5 = 0
= l n 2
x 0 , 5
Z a t e m x 0 , 2 5 = 0
- 0 . 5
- 1 0 - 8 - 6 - 4 - 2 0 2 4 6 8 1 0
" F ( x ) = 0 , 5 Ô! x = l n 2
Z a t e m x 0 , 5 = l n 2 H" 0 , 6 9 3 1
" F ( x ) = 0 , 7 5 Ô! 1 - e - x = 0 , 7 5 Ô! x = l n 4
Z a t e m x 0 , 7 5 = l n 4 H" 1 , 3 8 6 3
6
( e ) W y l i c z y - o i l e t o m o |l i w e - w a r t o [ o c z e k i w a n i w a r i a n c j o r a z w y z n a c z y m e d i a n i k w a r t y l e
ñø
ôø 0 d l a z 0 ,
òø
r o z k Ba d u z m i e n n e j l o s o w e j Z o d y s t r y b u a n c i e F ( z ) = 1 - ( 1 - z ) 2 d l a 0 < z 1 ,
ôø
óø
1 d l a z > 1 .
2 ( 1 - z ) d l a 0 < z < 1 ,
" J e s t t o r o z k Ba d c i g By o g s t o [c i f ( z ) = F ( z ) =
0 p o z a t y m .
ëø
1 öø
" 1
1
íøz 2 z 3 øø
" E Z = z f ( z ) d z = 2 z ( 1 - z ) d z = 2 - = H" 0 , 3 3 3 3
2 3 3
- " 0
0
ëø
1 öø
2 2
" 1
1 z 3 z 4 1 1
íø øø
" D 2 Z = z 2 f ( z ) d z - ( E Z ) 2 = 2 z 2 ( 1 - z ) d z - = 2 - - = H" 0 , 0 5 5 6
3 3 4 3 1 8
- " 0
0
"
( D 2 Z H" 0 , 2 3 5 7 )
"
" F ( z ) = q Ô! ( 1 - z ) 2 = 1 - q Ô! z = 1 - 1 - q d l a 0 < q < 1
" " "
" Z a t e m z 0 , 2 5 = 1 - 0 , 7 5 H" 0 , 1 3 4 0 , z 0 , 5 = 1 - 0 , 5 H" 0 , 2 9 2 9 , z 0 , 7 5 = 1 - 0 , 2 5 H" 0 , 5
P r z y k Ba d y d o z a d a n i a 9 . 3 :
( a ) P r o m i e D k u l i R m a r o z k Ba d j e d n o s t a j n y U ( 4 , 9 ; 5 , 1 ) c m . K u l w y k o n a n o z |e l a z a o g s t o [c i 7 , 8 8
g / c m 3 . W y l i c z y - o i l e t o m o |l i w e - w a r t o [ o c z e k i w a n i w a r i a n c j l o s o w e j m a s y M t e j k u l i
w y k o r z y s t u j c r o z k Ba d p r o m i e n i a l o s o w e g o R ( p a t r z t a k |e p r z y k Ba d 8 . 5 ( c ) ) .
" M a s a k u l i r ó w n a j e s t M = a - 3 R 3 , g d z i e a = ( 4 · 7 , 8 8 À/ 3 ) - 1 / 3 H" 0 , 3 1 1 7 .
0 , g d y r " [ 4 , 9 ; 5 , 1 ] ,
/
" G s t o [ R m a p o s t a : f R ( r ) =
1
= 5 , g d y r " [ 4 , 9 ; 5 , 1 ] .
5 , 1 - 4 , 9
5 , 1
"
" E M = a - 3 E R 3 = a - 3 r 3 f R ( r ) d r = 5 a - 3 r 3 d r = 5 a - 3 5 , 1 4 - 4 , 9 4 H" 4 1 2 7 , 6 g .
4
- " 4 , 9
"
" D 2 M = E M 2 - ( E M ) 2 = a - 6 E R 6 - ( E M ) 2 = a - 6 r 6 f R ( r ) d r - ( E M ) 2 =
- "
5 , 1 2
= 5 a - 3 r 6 d r - ( E M ) 2 = 5 a - 3 5 , 1 6 - 4 , 9 6 - 5 a - 3 5 , 1 4 - 4 , 9 4 H" 2 0 4 3 3 , 6 8 6 g 2 .
6 4
4 , 9
( b ) Z m i e n n a l o s o w a X m a r o z k Ba d C a u c h y e g o C ( 0 , 1 ) . W y l i c z y - o i l e t o m o |l i w e - w a r t o [ o c z e -
k i w a n i w a r i a n c j z m i e n n e j l o s o w e j Y = a r c t g X w y k o r z y s t u j c r o z k Ba d z m i e n n e j l o s o w e j X ,
( p a t r z t e | p r z y k Ba d 8 . 5 ( d ) ) .
1
" X m a g s t o [ p o s t a c i f X ( x ) = .
À( 1 + x 2 )
"
"
1 1
" E Y = E a r c t g X = a r c t g x f X ( x ) d x = a r c t g x · d x =
À x 2 + 1
- "
- "
ëø
0 "öø
1
íø
= a r c t g 2 x + a r c t g 2 x øø = 0 .
2 À
- " 0
"
2
" D 2 Y = E Y - ( E Y ) 2 = E a r c t g 2 X = a r c t g 2 x f X ( x ) d x =
- "
ëø öø
0
"
3
1 1 1 2 À À2
íø
= a r c t g 2 x d x = a r c t g 3 x + a r c t g 3 x "øø = · = H" 0 , 8 2 2 5 .
À x 2 + 1 3 À 3 À 2 1 2
- " 0
- "
7
P r z y k Ba d y d o z a d a n i a 9 . 4 :
( a ) B Bd p o m i a r u d Bu g o [c i [r u b y m a s t a n d a r d o w y r o z k Ba d n o r m a l n y . Z n a l e z p r a w d o p o d o b i e Ds t w o ,
|e b Bd z a w a r t y b d z i e w p r z e d z i a l e [ 0 ; 0 , 5 ] , [ - 1 ; 1 ] , [ - 2 , 3 ; 2 ] .
" O z n a c z m y b Bd p o m i a r u d Bu g o [c i [r u b y p r z e z B .
W i e m y , |e B t o z m i e n n a l o s o w a o s t a n d a r d o w y m r o z k Ba d z i e n o r m a l n y m .
P ( B < x ) = P ( B x ) = ¦( x ) .
W a r t o [c i f u n k c j i ¦ o d c z y t u j e m y z t a b l i c .
" P ( B " [ 0 ; 0 , 5 ] ) = P ( 0 B 0 , 5 ) = ¦( 0 , 5 ) - ¦( 0 ) = 0 , 6 9 1 5 - 0 , 5 = 0 , 1 9 1 5
" P ( B " [ - 1 ; 1 ] ) = P ( - 1 B 1 ) = ¦( 1 ) - ¦( - 1 ) = ¦( 1 ) - ( 1 - ¦( 1 ) ) = 2 ¦( 1 ) - 1 =
= 2 · 0 , 8 4 1 3 - 1 = 0 , 6 8 2 8
" P ( B " [ - 2 , 3 ; 2 ] ) = P ( - 2 , 3 B 2 ) = ¦( 2 ) - ¦( - 2 , 3 ) = ¦( 2 ) - ( 1 - ¦( 2 , 3 ) ) =
= 0 , 9 7 7 2 - ( 1 - 0 , 9 8 9 3 ) = 0 , 9 6 6 5
( b ) D Bu g o [ p r o d u k o w a n y c h d e t a l i m a r o z k Ba d N ( 0 , 9 ; 0 , 0 0 3 ) . N o r m a p r z e w i d u j e w y r o b y o w y m i a -
r a c h 0 , 9 ± 0 , 0 0 5 . J a k i p r o c e n t p r o d u k o w a n y c h d e t a l i n i e s p e Bn i a w y m o g ó w n o r m y ?
" O z n a c z m y d Bu g o [ p r o d u k o w a n y c h d e t a l i p r z e z L .
J e s t t o z m i e n n a l o s o w a o r o z k Ba d z i e N ( m = 0 , 9 ; Ã = 0 , 0 0 3 ) .
L - m
W i e m y , |e m a r o z k Ba d N ( 0 , 1 ) .
Ã
" D e t a l s p e Bn i a w y m o g i n o r m y , g d y 0 , 9 - 0 , 0 0 5 L 0 , 9 + 0 , 0 0 5 .
" P ( d e t a l n i e s p e Bn i a w y m o g ó w n o r m y ) = 1 - P ( 0 , 9 - 0 , 0 0 5 L 0 , 9 + 0 , 0 0 5 ) =
0 , 9 - 0 , 0 0 5 - 0 , 9 L - m 0 , 9 + 0 , 0 0 5 - 0 , 9 5
= 1 - P = 1 - ¦ - ¦ - 5 =
0 , 0 0 3 Ã 0 , 0 0 3 3 3
5
= 2 1 - ¦ H" 2 ( 1 - ¦( 1 , 6 7 ) ) = 2 ( 1 - 0 , 9 5 2 5 ) = 0 , 0 9 5 .
3
" O d p . 9 , 5 % p r o d u k o w a n y c h d e t a l i n i e s p e Bn i a w y m o g ó w n o r m y .
( c ) A s y s t e n t p r o w a d z c y z a j c i a z e s t a t y s t y k i p r z y c h o d z i d o s a l i n a o g ó B 2 m i n u t y p r z e d w y z n a -
c z o n g o d z i n r o z p o c z c i a z a j . Z a k Ba d a j c , |e c z a s p r z y j [c i a j e s t z m i e n n l o s o w o r o z k Ba d z i e
n o r m a l n y m z à = 2 m i n u t y , o k r e [l i , j a k i e j e s t p r a w d o p o d o b i e Ds t w o s p ó zn i e n i a s i t e g o a s y -
s t e n t a n a z a j c i a . ( C z y a s y s t e n t p o w i n i e n z m i e n i z w y c z a j e ? )
" P r z y j m i e m y , |e m o m e n t r o z p o c z c i a z a j t 0 = 0 .
O z n a c z m y p r z e z T m o m e n t p r z y j [c i a a s y s t e n t a .
J e s t t o z m i e n n a l o s o w a o r o z k Ba d z i e n o r m a l n y m N ( m = - 2 , Ã = 2 )
( m = - 2 , b o a s y s t e n t p r z y c h o d z i n a o g ó B 2 m i n u t y p r z e d c h w i l t 0 = 0 ) .
0 - ( - 2 )
T - m
" P ( a s y s t e n t s i s p ó zn i ) = P ( T > 0 ) = P > = 1 - ¦( 1 ) =
à 2
= 1 - 0 , 8 4 1 3 = 0 , 1 5 8 7 .
8
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