: . . , .
. , 1, 2, 3, 4, 5, 6. - { 2, 4, 6 }. 6 , , 5 , .
, , . , , , ().
. 40 , 17 . 17 / 40 0,425. ( ), , ½, , .
, , . . , 1, ( ) , , 0, ( ) .
, , :
( ) = () + () - ( ),
( ) - , .
, , , ( ) = 0. A A ( A ) , , (A A) 1, , p ( A) = 1 p (A)
, , .
. , , , 1, 2, 3, 4, 5, 6.
, , n , n , . : X, Y M (X + Y) = MX + MY. : M (a X) = a M (X).
, .
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n , n / 6 ,
(1 n / 6 + 2 n / 6 + 3 n / 6 +
4 n / 6 + 5 n / 6 + 6 n / 6) / n =
(1 + 2 + 3 + 4 + 5 + 6) / 6 = 3,5
, 3,5.
X , ( ) x 1, x 2,.... :
M (X) = x 1 p (X = x 1) + x 2 p (X = x 2) +...,
p (X = xk ) - , X xk.
. , ½ 0 1.
0 ½ + 1 ½ = ½.
g , X - . g (X) , :
M g (X) = g (x 1) p (X = x 1) + g (x 2 ) p (X = x 2) +...,
:
D = (| X - MX |2).
, X a
D (a X) = a 2 DX.
. , , .
, a , a. a, .
:
D = (| X - MX |2) = M (X X 2 (MX) X + (MX) (MX)) = M (X X) - 2 (MX) MX + (MX) (MX) = M (X 2) - (MX)2;
, X
M (X 2) - (MX)2.
. , , .
X FX (s), : FX (s) = p (X < s) s.
. X , ,
FX (s) = 0, s ≤ 0, FX (s) = ½, 0 < s ≤ 1 FX (s) = 1, s > 1.
. , .
, ] 0, 1 [. E. :
FE (s) = 0, s≤ 0, FE (s) = s, 0 < s < 1 FE (s) = 1, s ≥ 1.
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X , :
FX (s) = [ -∞, s ] fX (u) du
s. fX X. E 1 ] 0, 1 [ .
X fX
M (X) = [ -∞, +∞, ] s fX (s) dt.
g (X), g ̶ :
M (g (X)) = [ -∞, +∞ ] g (s) fX (s) ds
( g X, g (X) ̶ ; g s, g (s) ̶ ).
,
D (X) = M (| X MX |2) = [ -∞, +∞ ] | s MX |2 fX (s) ds;
M (X 2)) = [ -∞, +∞ ] s 2 fX (s) ds.
,
D (X) = M (X 2) - (MX)2
. E, ] 0, 1 [ , .
X = α E + β, E , ] 0, 1 [, α, β , α > 0. s :
X < s, α E + β < s, E < (s - β) / α,
, FX (s) = FE ((s - β) / α). α E + β ] β, α + β [.
. , , , ] a, b [, .
, N ( , ), :
fN (s) = exp (- (s - m)2 / (2 σ2)) / (σ (2 π)½).
m σ , m , σ .
. , , m, σ.
N m, σ, (N m) / σ (0, 1).
. , N m, σ , (N m) / σ (0, 1).
X 1, X 2,..., Xn , A 1, A 2,..., An , X 1 A 1, X 2 A 2,... Xn An, :
p ( X 1 A 1, X 2 A 2, ... Xn An) = p ( X 1 A 1 ) p ( X 2 A 2 )... p ( Xn An).
. X 1, X 2,..., Xn , m d. n , (X 1+ X 2 +...+ Xn) / n m d / n.
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, , , .