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. , . (;b).
1) .
2) .
( ) X- f(x)- F(X):
F(x)= (x).
.
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. F(x)=
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1) f(x) 0
2) P(x1 X <x2)=
3)
4_) P(x X <x + x) f(x) x, x-.
.
X [;b], :
M(X)= ;
D(X)= .
1. D(X)=
X x,
M(X)= .
D(X)= .
2. D(X)= .
δ()=
3 .
,
0, x£0
F(x)= x, 0<x£1
1, x>1
0, x£0
f(x)=F`(x)= x, 0<x£1
1, x>1
¥ 0 ¥ 0 1 ¥ 1 1
M(X)=òxf(x)dx=òxf(x)dx+òxf(x)dx=ò(x*0)dx+ò(x*1)dx+ò(x-0)dx=0+òxdx+0=x2/2ô=1/2
-¥ -¥ 0 -¥ 0 1 0 0
¥ 1 1
D(x)=òx2f(x)dx-[M(x)]2=òx2dx-1/4=x3/3ô-1/4=1/3-1/4=1/12.
-¥ 0 0
.
1) , , ,
.
, . .. .
(a;b)
0, x£a
f(x)= 1/(b-a), a<x£b
0, x>b
M(X)=(a+b)/2; D(X)=(b-a)2/12; δ(X)=(b-a)/2
(a;b) (0;1).
0, x£0
f(x)= 1, 0<x£1 M(X)=1/2; D(X)=1/12; δ(X)= =1/2 = /6
|
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0, x>1
2) - ,
f(x)= e-
M(X)=a; D(X)=δ2; δ(X)= =δ
a=0 δ=1 .
:
Φ(x)= e ( )
:
x x
F(x)=òf(x)dx= òe dz
-¥ -¥
:
x
F(x)= òe dz
-¥
F0(x): F0(x)+F0(-x)=1ÞF0(-x)=1-F0(x)
( )
.
1) D(f): xÎR
2) f(x)>0 "xÎR
3) lim f(x)=0 Þ 0-
x¥
4) fmax=f(a)=
5) =
6) : x=aδ; f(aδ)=
=0 δ=1 Φ(x)= e .
.
a (. ) , : , a , , a .
C , , .. ; .
.
- ( ).
.
. 30 10. , , (10;50).
.
a=30; =10.
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(*)
.
(*)
t=3,
, , .
: , , , , , , .
.
() , P(X<Me(X))=P(X>Me(X))= .
x=Me(X) . x=Me(X) , .. F(Me(X))= .
0() i f(x) .
( ), ( ) ( ) .
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. , () >0 :
, .
. - , ()=1(.) , , 5 .
.
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. , >0 .