Biometrija_2011
.pdf
21
II.
2.1.
, ,
, , -
. ,
, - ). . -
, -
. : 17, 12, 19, 10, 20, 16, 24, 19, 16, 6, 16, 12, 13, 15, 14, 15, 16, 18, 21, 17, 7, 23, 12, 14.
-
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.
, ( -
- . , -
). -
, -
. -
, ..
, -
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.
2.2.
, n
: X1, X2,.... Xn.
.
( X ) -
:
n
X =(X1 + X2 + ....+ Xi +..Xn )/n = X i / n . (2.1)
i 1
-
, . , n
X1 m1 - , X2 –m2 , ........, Xk – mk - ( , m1 + m2 +....+m k = n). -
:
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X =(m1*X1+m2*X2 +.....+m k*X n)/n =
n
= m1*X1/n + m2*X2/n+...+ m k*X k/n. = i X i
i 1
n ( n ), m1/n, m2/n,...., mk/n
X –p1 , p2 ,...., pk -
, -
.
:
n |
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M (x) lim i X i |
pi X i . |
(2.2) |
i 1 |
i 1 |
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, -
- ( ).
-
. -
.
, -
( ), :
(Xi - (x)) ( (Xi - X )).
.
,
, ,
. : (Xi - (x))2
(Xi- X )2.
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. -
. -
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D(x) lim i ( X i |
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pi ( X i M (x))2 |
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,
-
(Xi - X )2 :
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(2.3) |
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D (x) (X1 X)2 (X2 X)2 (Xn X)2 /n (Xi X)2 /n |
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D (x) : |
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D (x) m1 ( X1 |
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i ( X i |
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(2.4) |
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(2.3-2.4) |
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,
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. -
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SX / Y |
(1 0,5642 ) 40,388 4,5 |
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(20 2) 0,0756 |
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: 8, 10, 7, 15, 12, 18, 20, 16, 25, 20, 18, 19, 22, 21, 18, 24, 19, 16, 18, , 22, 12, 17, 16 18, 14, 15, 17, 19, 11, 14.
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,
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:
1) ( i )
; ;
2) , – mi
Xi;
3)(2.1) (2.2);
4)(Xi- X );
5)(Xi - X )2 ;
6)(2.3) (2.4);
7)(2.5).
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= 16,7 17; |
D (x) 18,5; |
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, (17±4)
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2.3.
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(pi . -
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1 ( ,
n=30). X(min)=7 X(max)=25 X
; m =0 p =0 ( . 2)
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2.1).
, - , 30 -
7, 8, ... 25 .
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2 ( )
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0,099 |
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p(x)
0.16 |
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26 X |
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. 2.1 |
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. . 2.2 , -
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n X
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2.3.1.
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, -
, , . -
m, , n
. , 3 10 , 7 15
. , m - , 0 mi
n.
,
n m .
,
p. , ( -
), (1-p). , n -
m , (n-m) . -
26
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pn*(m)=p( )*p(A)*...*p(A)*p( |
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pn*(m) = pm * (1-p)n-m . |
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, ,
A . ( ),
, pn*(m).
, , -
n m . ,
n ( ). n -
a,b,c...... . (a , b- , -
.),
C = n!/( !* !* !).
, "!" - - -
1 "!". , 5!= 1*2*3*4*5. -
0! 1. m
( ) (n-m) (1- ).
( )
= n!/ [m!*(n-m)!].
(a+b)n, n(m) n
m.
,
m n -
( ) (m) .
( " "): A , -
, .
p(m) = p*(m) + p*(m) + |
.......+p*(m) = n(m)*pn*(m). |
( ), -
pn(m) = n(m) * pm * (1-p)n-m |
(2.6) |
, m - n -
0 n: 0 m n.
(2.6):
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pn(0) = Cn(0)* p0 * (1-p)n ; pn(1) = Cn(1)* p1 * (1-p)n-1
pn(2) = Cn(2)* p2 * (1-p)n-2
.............................
pn(n) = Cn(n)* pn * (1-p)0 .
;
;
n(m)
, , (2.6), -
( ).
m,
) n , .
m, ,
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(m) = n * p . |
(2.7) |
m -
:
D(m) = n * p * (1 - p). |
(2.8) |
. m
20 (0 m 20 ).
0,5. . 2.2.
, 10 .
10.
(2.7).
0,7. , 6
(m)
0.2
0.1
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. 2.2 |
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4 . ,
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p=0,7 |
p6(4) = 6(4)*p4*(1-p)2 = |
n=6 |
= [(1*2*3*4*5*6)/(1*2*3*4*1*2)]*(0,7)4*(0,3)2 = 0,324 |
m=4 |
p(4)-? (m) = 6*0,7 = 4,2 4;- ?
D(4)-? D(m) = 6*0,7*0,3 = 1,26; (m) = D(m) =1,12 1.(m)-?
(m) (m) , m
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2.3.2.
n p -
. ,
, 50 , -
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p60(30) =((50!)/(30! * 20!))*(0,5)50.
, 20! 19 .
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pn (m) |
1 e (m M )2 /(2 D) . |
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2 D |
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n (p<<1) -
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.
. 2.3 -
. n=30, p=0,3; -
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n=30, p=0,05, -
=1,5. , -
, 30 ,
0,223.
1) |
2) |
. 2.3
2.4.
-
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X1= X2= + X = b
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0 
X
X 
b
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p/ X. -
(X), p/ X, X
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X.
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f(x) Lim( p / x)
0
30
f( ) .
f( ) , -
p( x):
f (x) dp dx
f( ) , -
X X1 = X2 = b
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b
p(a X b) f (x)dx
a
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M (x) X f (x)dx . |
(2.11) |
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D(x) ( X M (x)) 2 f (x)dx . |
(2.12) |
(x) D(x) .
f( ) -
( . 2.5).
. 2 .5