(.4634).
.
1. .
(, , ) , - .
, , , , 5 , , 4,5 5,4. , 5,0 , , 4,95 5,04 , , 5,00 , , 4,995 5,004.
=5,15 10%, 5,15 , 5,145 5,154 5,150,05 0,05/5,15=0,0097≈0,01=1% 5,151%.
10%, 5,20,5, , 4,7 5,7 ( 0,5/5,15=10%).
σ=22,15 10%, 22,15 , σ 22,15+0,01 22,15-0,01 (σ=0,045%) , 0,05, 22,15+0,05 22,15-0,05 (σ=0,23%). 20 10%, , σ 20-2 20+2, .. 22,15 .
σ=22 10%, , σ 22-2,2 22+2,2. σ=22 , 22-1 22+1, 5%, 10%.
, , , , , , (0,045-0,23% 10%.).
, , .
, , . , . .
() m, , n: m/n. n , .
, :
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(1+2+3+...+n)=(1)+(2)+ (3)+...+(n)
:
(1*2*3*...*n)=(1)*(2)*p(3)*...*(n)
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1. , . . 0,8, 0,9 0,8. () .
=0,8*0,9*0,8=0,576
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1 (Si=1), .. .
. , i i.
.
F(X).
, :
x
F(x)={1/[sx (2p)1/2 ]} ò[exp(-(x-mx )2/2sx2 ] dx
-¥
mx
mx = Σx/N,
N- , -¥ <x<+¥, σ2 ,
σ2= Σ(mx )2)/(N-1)
.
.
Fo(x)={1/[ (2p)1/2 ]} ò[exp(-(x)2/2 ] dx
-¥
(1 <x<x2 )=Fo(x2)- Fo(x1)
Fo(x)-(1/2) 0 4. () 0 1.
, (), :
=(x-mx)/ sx
.
, Fo(x2)- Fo(x1) = (01) - (02)
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<Xm:
(m<)=F(x).
F(x) .
(1<<2) = F(x2)-F(x1)
F(-¥)=0; F(+¥)=1.
f(x)=dF(x)/dx
.
( ) . k- :
mk= Sxik pi
¥
m=ò xF(x)dx
-¥
( ) .
m1=Sxi pi
, ( ) - ( ).
.
(D). :
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n
D(x)=M[(x-mx )2] = S((xi-mx )2pi
i=1
pi=ni/N (N - , ni - , i) , ,
¥
D(x) = ò ((x-mx )2f(x)dx
-¥
Dx. Dx=s2 . s .
, s3 :
g = m3/ s3
, :
m3 =m3-3m1m2 +2m13
()
t=(m4/s 4)-3
, -2< t < +¥. t>0 - , , t< 0 - . t= - 2 .
1- 2- ,
=(-mx)/s
()=0; D(xo )=1.
,
f(x)=(1/sx )f1(xo )=(1/sx)f1 [(X-mx )/sx]
f1(xo)= sxf(x)= sxf(mx +sxxo)
F(x)=F1(xo)= F1 [(x-mx )/ sx]
()
D =S/n1/2 ; Ds =S/(2N)1/2;; Dg =(6/N)1/2 ; Dt =2Dg
S- .
, :
g£2Dg t£2Dt =4Da
|g1 | £3(D(g1))1/2 t £=5(D(t))1/2
D(g1 )=6(n-1)/[(n+2)(n+3)];
D(t)=24n(n-2)(n-3)/[(n+1)2 (n+3)(n+5)]
e=(D()/ ).100%
g1 t . 1. g1 >0, . g1<0, . t>0, . t<0, .
3-4 , .
, , , .
, , N . =Σ/N , .(. .1).
.1. ( ) .
.2. .
(. .2), (), , () . (lg).
. σ =Σ(-)2/(N-1) , , 72% +σ -σ. , , , 95%, - ε +ε. ε σ t:
ε=σt/(N)0,5.
t 1, t N c .
1. t: t N ( ) .
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N-1 | = 0,90 | P= =0,95 | P= =0,98 | = 0,99 | N-1 | = 0,90 | P= =0,95 | P= =0,98 | = =0,99 |
6,31 | 12,704 | 31,821 | 63,7 | 1,80 | 2,201 | 2,718 | 3,11 | ||
2,92 | 4,303 | 6,965 | 9,93 | 1,78 | 2,179 | 2,681 | 3,06 | ||
2,35 | 3,182 | 4,541 | 5,84 | 1,77 | 2,160 | 2,650 | 3,01 | ||
2,13 | 2,778 | 3,747 | 4,60 | 1,76 | 2,145 | 2,624 | 2,98 | ||
2,02 | 2,571 | 3,365 | 4,03 | 1,75 | 2,131 | 2,602 | 2,95 | ||
1,94 | 2,447 | 3,143 | 3,71 | 1,75 | 2,120 | 2,583 | 2,92 | ||
1,90 | 2,365 | 2,998 | 3,50 | 1,74 | 2,110 | 2,567 | 2,90 | ||
1,86 | 2,306 | 2,896 | 3,36 | 1,73 | 2,086 | 2,528 | 2,85 | ||
1,83 | 2,262 | 2,821 | 3,25 | 1,71 | 2,060 | 2,485 | 2,79 | ||
1,81 | 2,288 | 2,764 | 3,17 | 1,70 | 2,042 | 2,457 | 2,75 |
, (.. ε) N.
σ ε, , f = 22 /2. , , f 21,5 22,4 σ 22-0,5 22+0,5 ε ≈ 0,5. 0,5/22*100%= 2,3%, . +-15%. , f , 20, 2.101, , 15 25 20-5 20+5, ε=5 ε=5/20≈25%.
2.
.
, D= êx-m ê e
(D£e)=(mx-e £ mx+e)
P(-e£ xo £ e)=(e)-(-e)=2(e)
, mx s
(D£e)=(D£e/s)=2(e/s)
k=e/s :
(D£ks)= 2(k)
P(D£s)=2(1)=0,6826
P(D£2s)=2(2)=0,9544
P(D£3s)=2(3)=0,9973
.
3. h , s =0,3 . , Dh³0,5 ?
: (Dh£0,5)=2(0,5/0,3)=0,905
(Dh³0,5)=1-0,905=0,095.
.
(n ) (..
. k . k:
k=1+3,2lg(n)
h=(X -X)/k.
, n. i- i =ni/n. , i-
Xi* =(Xi-1 +Xi)/2
-,
Xi * ni. , - . . , ni , ( X).
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.
Xcp =S Xi /n S2=S(Xi -Xcp)2 /n. S .
n/(n-1). :
S2 =S(Xi -Xcp)2/(n-1). :
S2=│(SXi 2 -(SXi)2)/n│
.
, .
- , , , .
, X n , , X s2. (X -Xi)/sn-1/2) , . s2
. () , S2 ( ) t:
t=((Xcp - mx)/S) n1/2
t , n. f(t) - ¥ t Sn(t) to , t:
Sn(t)=[t <t]
S(t) - t-. , ( ).
(U-Ucp)^2 | ||||||
5,1 | 26,01 | |||||
0,1 | 0,01 | |||||
-1,9 | 3,61 | |||||
3,1 | 9,61 | |||||
2,1 | 4,41 | |||||
2,1 | 4,41 | |||||
4,1 | 16,81 | |||||
1,1 | 1,21 | |||||
0,1 | 0,01 | |||||
-0,9 | 0,81 | |||||
summa | 66,9 | |||||
srednee | 22,9 | |||||
σ^2= | =66,9/ (10-1) | 7,433333 | ||||
σ= | 2,726414 | |||||
ε= | 2,262*S/N^0,5 | |||||
1,950224 | ||||||
U= | 22,9+-1,95 | |||||
U≈ | 23 2 | |||||