→ 1, 2, 3
P (X = 1) =
3
5
= 0, 6; P (X = 2) =
2
5
·
3
4
= 0, 3; P (X = 3) =
2
5
·
1
4
.1 = 0, 1
→ X = 0; 1; 2
H
i
P (H
0
) =
C
0
8
.C
2
2
C
2

10
=
28
45
H
0
, H
1
, H
2
P (X = 0|H
0
) =
C
0
7
C
2
5
C
2
12
=
10
66
; P (X = 0|H
1
) =
C
0

i
)P (X = 0|H
i
) =
1
45
·
10
66
+
16
45
·
6
66
+
28
45
·
3
66
=
190
2970
= 0, 06397
P (X = 1|H
0
) =
C
1

9
C
1
3
C
2
12
=
27
66
⇒ P (X = 1) =
2

i=0
P (H
i
)P (X = 1|H
i
) =
1
45
·
35
66
+
16
45
·
32
66

1 2 < x
E(X) = −5.0, 4 + 2.0, 3 + 3.0, 1 + 4.0, 2 = −0, 3
V (X) = (−5)
2
.0, 4 + 2
2
.0, 3 + 3
2
.0, 1 + 4
2
.0, 2 − (−0, 3)
2
= 15, 21
σ
X
=

V (X) =
√
15, 21 = 3, 9
m
0
⇒ m
0
= −5
E(X) = 0.0, 05+1.0, 12+2.0, 17+3.0, 08+4.0, 12+5.0, 2+6.0, 07+7.0, 02+8.0, 07
+9.0, 02 + 10.0, 03 + 11.0, 05
= 4, 33
V (X) = 0
2

V (X) =
√
8, 3411 = 2, 8881
σ
X
E(
X) =
E(X
1
) + E(X
2
) + E(X
3
)
3
= 0, 8
V (
X) =
V (X
1
) + V (X
2
) + V (X
3
)
9
= 0, 12
E(X) = 20.0, 2 + 25.0, 3 + 30.0, 15 + 35.0, 1 + 40.0, 25 = 29, 5
V (X) = 20
2

.3 −
2
5
.2 = 1
V (Z) =

3
5

2
V (X) +

2
5

2
V (Y ) =

3
5

2
.3 +

2
5

2
.2 =
35

2
= 0, 69
F (x) =





















0 x  2
0, 192 2 < x  3
0, 432 3 < x  4
0, 74 4 < x  5
1 x > 5
E(X) = 2.0, 192 + 3.0, 231 + 4.0, 308 + 5.0, 269 = 3, 654

1
+ P
3
= 0, 1
0, 9 = E(X
2
) = (−1)
2
.P
1
+ 0
2
.P
2
+ 1
2
.P
3
⇒ P
1
+ P
3
= 0, 9
⇒ P
1
= 0, 4; P
3
= 0, 5; P
2
= 1 − 0, 4 − 0, 5 = 0, 1

2
= 3
→ X = 0; 1
A
i
P (X = 0) = P (
A
1
A
2
) = P (A
1
)P (A
2
/A
1
) =
4
5
·
3
4
= 0, 6
P (X = 1) = 1 − P (X = 0) = 1 − 0, 6 = 0, 4
→ E(X) = 0.0, 6 + 1.0, 4 = 0, 4
V (X) = 0
2
.0, 6 + 1
2
.0, 4 − 0, 4

=

V (Y ) =
√
180, 29 = 13, 427

P
x
= 0, 1 + 0, 2 + 0, 35 + 0, 2 + 0, 1 + 0, 05 = 1
E(X) = 0.0, 1 + 10.0, 2 + 20.0, 35 + 30.0, 2 + 40.0, 1 + 50.0, 05 = 21, 5
P (X > 20) = P (X = 30) +P (X = 40)+ P (X = 50) = 0, 2 +0, 1+ 0, 05 = 0, 35
→ X = 2, 3, , 12
1
36
2
36
3
36
4
36
5
36
6
36
5
36
4
36
3
36



























0 ; x  2
1
36
; 2 < x  3

; 11 < x  12
1 ; x > 12
m
0
= 7
P (X  12) = P (X = 12) + P (X = 13) + P (X = 14) + P (X = 15) =
0, 2 + 0, 15 + 0, 1 + 0, 05 = 0, 5
E(X) = 9.0, 05+10.0, 15+11.0, 3+12.0, 2+13.0, 15+14.0, 1+15.0, 05 = 11, 75
V (X) = 9
2
.0, 05 + 10
2
.0, 15 + 11
2
.0, 3 + 12
2
.0, 2 + 13
2
.0, 15 + 14
2
.0, 1 + 15
2
.0, 05 −
− 11, 75
2
= 2, 2875
σ
X
=


3
) = 0, 2.0, 4.0, 5 = 0, 04
P (X = 1) = 0, 2.0, 6.0, 5 + 0, 8.0, 4.0, 5 + 0, 8.0, 6.0, 5 = 0, 46
P (X = 2) = 1 − 0, 24 − 0, 46 − 0, 04 = 0, 26
→ Y = 3X
E(Y ) = 3E(X) = 3(0.0, 24 + 1.0, 46 + 2.0, 26 + 3.0, 04) = 3, 3
E(X) = 1.0, 3 + 4.0, 1 + 8.0, 6 = 5, 5
P (|X − E(X)| < 4) = P (|X − 5, 5| < 4) = P (5, 5 − 4 < X < 5, 5 + 4)
= P (1, 5 < X < 9, 5) = P (X = 4) + P(X = 8) = 0, 1 + 0, 6 = 0, 7
f(x) =



k(30 − x); x ∈ (0; 30)
0 ; x /∈ (0; 30)
1 =
+∞

−∞
f(x)dx = k
30

0
(30 − x)dx = k

30x −
x
2
2




12
0
= 0, 64
E(X) =
+∞

−∞
xf(x)dx =
30

0
x
1
450
(30 − x)dx =
1
450

15x
2
−
x
3
3





− 6x + 2 ; x ∈ (0; 1)
1 =
+∞

−∞
f(x)dx =
1

0
(3ax
2
− 6x + 2)dx = (ax
3
− 3x
2
+ 2x)


1
0
= a − 1 ⇒ a = 2
E(X) =
+∞

−∞
xf(x)dx =
1

0
x(6x


1
2
+
1
π
arctg1

−

1
2
+
1
π
arctg0

=
1
π
(
π
4
−0) =
1
4
f(x) = F
′
(x) =
1

1 ; x > 4
P (X < 3) = F (3) = (1/2).3 − 1 = 1/2
P (2  X < 3) = F (3) −F(2) = 1/2 − 0 = 1/2
F (x) =









0 ; x  0
sin2x ; 0 < x  π/4
1 ; x > π/4
f(x) = F
′
(x) =



0 ; x /∈ (0; π/4)
2cos2x ; x ∈ (0; π/4)
f(x) =



acosx ; x ∈ (−π/2; π/2)
0 ; x /∈ (−π/2; π/2)

√
2/4
E(X) =
+∞

−∞
xf(x)dx =
π/2

−π/2
(1/2)xcosxdx = (1/2)xsinx




π/2
−π/2
−(1/2)
π/2

−π/2
sinxdx =
= 0 + cosx




π/2
−π/2
= 0

xf(x)dx =
π

0
x.(1/2).sinxdx = (1/2)(−xcosx + sinx)




π
0
= π/2
F (x) =



1 − (x
0
/x)
α
; x  x
0
0 ; x < x
0
; α > 0
P (X > a) = 0, 5 ⇔ P (X  a) = 1 −P(X > a) = 1 − 0, 5 = 0, 5
⇔ F (a) = 0, 5 ⇔ 1 − (x
0
/a)
α

0
1 + cos2x
2
dx =
2
π

x
2
+
1
4
sin2x





π
4
0
=
1
4
+
1
2π
(0; π/4)
P () = C
2

−∞
f(x)dx =
3

0
(x
2
/9)dx =
x
3
27




3
0
= 1
P (1 < X < 2) =
2

1
f(x)dx =
2

1
x
2
9
dx =

0 = F (−∞) = c + b.
−π
2
1 = F (+∞) = c + b.
π
2
⇔



c = 1/2
b = 1/π
f(x) = F
′
(x) =
1
π
.
1
1 +

x
a

2
=
a
2
π(x
2

20
dx =
1
40
x
2




25
5
= 15
V (X) =
25

5
x
2
f(x)dx −(E(X))
2
=
25

5
x
2
1
20
dx − 15

a
= k(b − a) ⇒ k =
1
b − a
> 0
E(X) =
+∞

−∞
xf(x)dx =
b

a
x
1
b − a
dx =
1
b − a
·
x
2
2


b
a
=
a + b
2

b
a
−
(a + b)
2
4
=
(b − a)
2
12
F (x) =









0 ; x  a
x − a
b − a
; x ∈ (a; b)
1 ; x  b
f(x) =



Csin2x ; x ∈ (0; π/2)

e
x
+ e
−x
 0, ∀x ⇒ k  0
1 =
+∞

−∞
f(x)dx =
+∞

−∞
k
e
x
+ e
−x
dx = k
+∞

−∞
d(e
x
)
e
2x
+ 1
= karctg (e
x

2
> 0, 8)
P (X
i
> 0, 8) = P (X > 0, 8) = P (X  0, 8) − P (X = 0, 8) = 0, 85 −0, 25 = 0, 6
P (X
1
> 0, 8; X
2
> 0, 8) = P (X
1
> 0, 8).P (X
2
> 0, 8) = 0, 6.0, 6 = 0, 36
m
0
= 2
E(X) = −2.0, 1 − 1.0, 1 + 0.0, 2 + 1.0, 2 + 2.0, 3 + 3.0, 1 = 0, 8 > 0
V (X) σ
X
V (X) = (−2)
2
.0, 1 + (−1)
2
.0, 1 + 0
2
.0, 2 + 1
2
.0, 2 + 2
2

=

V (X
A
) =
√
150400 = 387, 81
E(X
B
) = −200.0, 1 + 50.0, 6 + 100.0, 3 = 40
V (X
B
) = (−200)
2
.0, 1 + 50
2
.0, 6 + 100
2
.0, 3 − 40
2
= 6900
σ
B
=

V (X
B
) =
√
6900 = 83, 06

.100% =




387, 81
60




.100% = 646, 36%
CV
B
=




σ
B
E(X
B
)




.100% =



























0 ; x  2
0, 2 ; 2 < x  3
0, 4 ; 3 < x  4
0, 7 ; 4 < x  5
0, 8 ; 5 < x  6

P (X  4) = P (X = 4) + P(X = 5) = 0, 3 + 0, 1 = 0, 4
Y =

E(X).3
E(Y ) = 3E(

E(X)) = 3(0.0, 1+1.0, 1+
√
2.0, 2+
√
3.0, 2+2.0, 3+
√
5.0, 1) = 4, 66
E(X) = 400.0, 05 + 500.0, 15 +600.0, 41+ 700.0, 34 +800.0, 04 + 900.0, 01 = 620
V (X) = 400
2
.0, 05 + 500
2
.0, 15 + 600
2
.0, 41 + 700
2
.0, 34 + 800
2
.0, 04 + 900
2
.0, 01−
−620
2
= 9000

0 ; x /∈ [4; 14]
1
14 − 4
; x ∈ [4; 14]
⇒ P (X > 8) =
+∞

8
f(x)dx =
14

8
1
10
dx =
x
10




14
8
= 0, 6
LN
i,j
=





0
x
n
e
−x
dx = n!
E(X) =
+∞

−∞
xf(x)dx =
1
m!
+∞

0
x
m+1
e
−x
dx =
1
m!
(m + 1)! = m + 1
V (X) =
+∞

−∞
x

x
3
; x  x
0
0 ; x < x
0
; x
0
> 0
⇒ f(x) = F
′
(x) =





3x
3
0
x
4
; x  x
0
0 ; x < x
0
E(X) =
+∞

−∞


−∞
x
2
f(x)dx −[E(X)]
2
=
+∞

x
0
x
2
3x
3
0
x
4
dx −
9x
2
0
4
= −
3x
3
0
x



3
) = P (A
1
)P (A
1
/A
2
)P (A
3
/A
1
A
2
) =
3
5
·
2
4
·
1
3
= 0, 1
⇒ P (A) = 1 − 0, 1 = 0, 9
→ X = 0; 1.2
P (X = 0) = P (
A) = 0, 1
P (X = 2) = P (A
1
A

)P (A
2
/A
1
)P (A
3
/A
1
A
2
)
+P (
A
1
)P (A
2
/A
1
)P (A
3
/A
1
A
2
)
=
2
5
·
1

0 ; x  100
100/x
2
; x > 100
P (X  150) =
150

−∞
f(x)dx =
150

100
100
x
2
dx = −
100
x




150
100
=
1
3
⇒
A
P (A) = 1/3; P (

−x/100




+∞
0
= 100k ⇒ k = 1/100
P (50 < X < 150) =
100

50
f(x)dx =
150

50
1
100
e
−x/100
dx = −e
−x/100


150
50
= e
−1/2
− e
−3/2

A
) = 20
2
.0, 3 + 80
2
.0, 5 + 120
2
.0, 2 − 70
2
= 1300
V (X
B
) = (−30)
2
.0, 3 + 100
2
.0, 5 + 140
2
.0, 2 − 69
2
= 4429
• X
1
→ X
1
= 0; 5; 10
P (X
1
= 0) = 0, 25.0, 25 = 0, 0625
P (X

E(Z) = 750.0, 03 + 1050.0, 09 + 1350.0, 12 + 1650.0, 15 + 1950.0, 22 + 2250.0, 21+
+2400.0, 18
= 1860
•
⇒ T =



30X − 4000 ; X  200
30X − 4000 − 25(X − 200) = 5X + 1000 ; X > 200
⇒ E(T ) = 1550.0, 03+1850.0, 09+2025.0, 12+2075.0, 15+2112, 5.0, 11+2125.0, 5
= 2062, 125
•
⇒ S =



30X − 4800 ; X  240
30X − 4800 − 25(X − 240) = 5X + 1200 ; X > 240
E(S) = 750.0, 03 + 1050.0, 09 + 1350.0, 12 + 1650.0, 15 + 1950.0, 22 + 2250.0, 21+
+2425.0, 13 + 2475.0, 05
= 1867
⌣
f(x) =





1




a
−a
= 1
F (x) =









0 ; x  −2
1
2
+
1
π
arcsin
x
2
; −2 < x  2
1 ; x > 2
P (−1 < X < 1) = F (1) −F(−1) = [
1
2

(x) =





0 ; x /∈ (−2; 2)
1
π
√
4 − x
2
; x ∈ (−2; 2)
n

i=1
m
i
k
i
n

i=1
m
i
k
i
N
=
a