: . , . , ()= 0,27; , ()= 0,7. =+, . (3) ()=(+)=()+(). , ()= 0,27+0,7=0,97.
4. , . , , 0,8, 0,7. , .
: , , , . , ( ), ( ), , . , =+, . (4) : ()=(+)=()+()- (). ()= 0,8, ()= 0,7. , ()=()() . , ()= 0,8+0,7-0,80,7=0,94.
5. , 20 25 . 2 . , .
: : ; . , (1) , = 25 , m=20 - , . () = (1) , , . n = 24, ; m = 19, , , .
P(B/A) = . (5): () = P(A)P(A/B). , () = .
6. 25% I, 30% II, III. I 1% , II 2% , III 1,5%. , . , I?
|
|
: : . : 1 I; 2 II, 3 III. (1) = 0,25; (2) = 0,30; (3) = 1- (0,25 + 0,30) = 0,45. : (1)+ (2)+ (3) = = 1, 1, 2, 3 .
: (/1) = 0,01; (/2) = 0,02; (/3) = 0,015. : () = (1) (/1) + (2) (/2)+ (3) (/3), ()= 0,250,01 + 0,300,02 + 0,450,015 = 0,01525 0,015.
, I, :
(1/A) = , (1/A)=
, 16% I.
3.
.. , 5, 1-3; 6, 5.
, Ā, , , (Ā) = 1 = q, .
:
, | (8) |
m= 0,1,2,..., n , n m .
7. 10 , 4 . , , 7 . 5 ? , ?
: n= 10. p= , 4 , . 5 10 (10), . n= 10, = 5, p= , q= 1 , q= ,
10 (5) = = = ≈ 0,0584.
, , =7+8+9+10= i, i i . .
()=(7)+(8)+(9)+(10)=710 7 q3+810 8 q2+ 910 9 q1+ 1010 10 q0 = ()7()3+ ()8()2+ ()9()1+ ()10()0= + + + = (41034+534+103+1) ≈0,0035.
.. . 6, 3-5; 7, 1-5; 8, 1-8; 10, 1-3; 11, 1-5.
: , , , .
X,Y,Z,, x,y,z,. .
, .
. : ; ; . : F(), f ().
|
|
:
F()= ( <), (9)
f (x) = F'() (10)
, (11)
P(α < < β)=F(β)-F() (12)
(α<<β)= f (x)dx, (13)
.
( ) :
()=1 1 +2 2 + + n pn, (14)
( )
D(x)=M(x2)- (15)
σ() = . (16)
:
()= f(x)dx, (17)
( )
D(X)= x2f(x)dx- | , (18) |
c
σ()= . (19)
8. . . , F(X) . (), D(X), σ().
: ( ) = , , q q= 1 , q= 1 = . , , , . , : 0= 0; 1= 1; 2= 2; 3 =3.
(10) = 0,1,2,3.
(= 0 )=3( 0 )=03 0q3=()3= ;
(= 1 )=3( 1 )=13 1q2 = ()1()2= ;
(= 2 )=3(2)=23 2q1 = ()2 = ;
(= 3 )=3( 3 )=33 3q0 = ()3= ;
pi |
xi |
1/8 |
3/8 |
1
xi | ||||
pi |
.1.
pi= + + + = 1( ).
(.1).
F(X):
1. < ≤0, F(x)= 0,
2. 0< x ≤1, F(x)= P(X=xi)=P(X= 0 )= ,
3. 1< x ≤2, F(x)= P(X=xi)= Pi= + = ,
4. 2< x ≤3, F(x)= P(X=xi)= Pi= + + = ,
5. 3< x≤ +∞, F(x)= P(X=xi)= Pi= + + + =1.
F(x) |
0 |
1 |
2 |
3 |
7/8 |
1/2 |
1/8 |
0 -∞ <x≤ 0;
0 <x≤ 1;
F(x)= 1 <x≤ 2;
2 <x≤ 3;
1 3 <x≤ +∞.
. 2.
(. 2).
():
()= xi pi= 0 +1 +2 +3 = .
(2):
(2)= xi2 pi =02 +12 +22 +32 =3.
()=(2) = 3 ()2= ,
: σ()= = = ≈ 0,87.
9. .
: ) , (0;1); ) f(x), F(x), f(x); ) (), () σ().
: ) (α<X<β)=F(β)-F(α), (0<X<1)= F(1)F(0)= .
) f(x) = F'(x) : .
|
|
F(x) |
x |
1 |
P(0<x<1) |
f(x) |
1 |
1/3 |
.3. F(x) .4. f(x)
) :
()= .()= = + + = = = = .
()
()= (2)- .
(2):
(2)= = + + = = =
= =1.
()= (2) =1- .
σ() :
σ()= , σ()= .