, . , .. .
. :
- ( );
- χ2;
- ;
- .
, .
( ) . .
, :
(14)
,
(15)
, , . .1 2.
0 m σ m m + σ x
.1. .
0 m x
.2. .
(1) (2), m σ . m = M (X),
σ = σ (), .. D (X) = σ2, π = 3,14159, e = 2,71828.
M (X) = m
σ () = σ, :
~ N (m, σ) ~ N (m, σ2).
, m = 0 σ = 1. () .
U (U ~ N (0,1)), ,
; (16)
f (u), F (u) , ( u). :
(17)
( ~ N (m, σ)) :
(18)
, ~ N (m, σ), ~ N(0,1).
, . (.1) = m. m σ. , σ, .
, :
|
|
( M () < σ) = 0,68; ( M () < 2σ) = 0,95;
( M () < 3σ) = 0,9973.
,
( ~ N (m, σ)) 99,73 % [ m 3σ, m + 3σ ].
, . , ~ N (m, σ) Y ~ N (my, σy) ,
Z = aX + bY ~ N (mz, σ), (19)
mz = a m + b my; σz 2 = a2 σx 2 + b2 σy 2.
. , , , .. .
( , ..).
, . . χ2 , .
χ2 ( )
i, i = 1, 2, , n mi σi , .. i ~ N (mi, σi).
, i = 1, 2, , n, , , Ui ~ N (0,1).
χ2 n (χ2 ~ χn2), (20).
, ( v) , , .
, , n , m , v = m n. , U2 ~ χ12.
(20) , χ2 v.
, χ2 , (.3). χ2 :
. 3. , χ2 .
M (χ2) = v = n m,
D (χ2) = 2 v = 2 (n m).
Y χ2 n k ( ~ χn2, Y ~ χk2), ( + Y) χ2 v = n + k.
χ2 . χ2 .
U ~ N (0,1), V U , χ2 n .
(21)
(t ) n
|
|
(T ~ Tn).
(21) , n . , , ( ) (.4)
. 4. , .
M (T) = 0,
D (T) = n / (n 2).
,
n > 30 .
, . .
V W , χ2 v1 = m v2 = n .
(22)
v1 = m v2 = n (F ~ Fm,n). , F m n.
m n (.5). , Tn2 = F1,n, Tn , v = n, F1,n , v1 = 1 v2 = n.
.5. , .
. .