A
P (A) = 0, 51
X ∼ B(n = 5; p = 0, 51)
P (X = 2) = C
2
5
.0, 51
2
.0, 49
3
= 0, 306
P (X  2) = P (X = 0) + P (X = 1) + P (X = 2) = C
0
5
.0, 49
5
+ C
1
5
.0, 51.0, 49
4
+
0, 306 = 0, 481
X ∼ B(n = 12; p = 1/3)
X ∼ B(n = 12; p = 1/3)
P (X = 4) = C
4
12
(1/3)
4
.(2/3)

V (X) = npq = 10000.0, 85.0, 15 = 1275
2
X ∼ B(n = 855; p = 0, 02)
m
0
np+p−1  m
0
 np+p ⇔ 855.0, 02+0, 02−1  m
0
 855.0, 02+0, 02 ⇔ m
0
= 17
⇒ X ∼ B(n = 5; p = 3/4)
P (X = 3) = C
3
5
.(3/4)
3
.(1/4)
2
= 0, 263
P (X  1) = 1 − P (X = 0) = 1 − C
0
5
.(1/4)
5
= 0, 999902
P (X  2) =

2

3
.0, 7
9
= 0, 2397
X ∼ B(n = 3; p = 0, 7)
P () = P (X = 2) + P (X = 3) = C
2
3
.0, 7
2
.0, 3 + C
3
3
.0, 7
3
= 0, 784
→ X ∼ B(n = 15; p = 0, 7)
P (X  10) =
15

i=10
C
i
15
.0, 7
i
.0, 3
15−i
= 0, 72162
P (X = x) =

5
20
+
C
5
12
C
0
8
C
5
20
= 0, 703818
⇒ X ∼ M(N = 20; M = 5; n = 3)
E(X) = n
M
N
= 3.
5
20
= 0, 75
Y = 2.50.X = 100X
E(Y ) = 100E(X) = 100.0, 75 = 75
→ X ∼ M(N = 20; M = 3; n = 5)
P (X = x) =
C
x
3
C
5−x

C
5
100
= 0, 23228
X ∼ B(n = 100; p = 0, 02)
E(X) = np = 100.0, 02 = 2
X ∼ P (λ)
E(X) = λ = 8
P (X > 4) = 1−P (X  4) = 1−e
−8
8
0
0!
−e
−8
8
1
1!
−e
−8
8
2
2!
−e
−8
8
3
3!
−e
−8

P (X = i) =
6

i=0
e
−2
2
i
i!
= 0, 9947
P
b
= P (X  12) = 1 − P (X < 12) = 1 −
12

i=0
e
−2
2
i
i!
= 0, 0000013646
P (X > 7) > 0, 1
P (X > 7) = 1−P (X  7) = 1−P (X  6)−P (X = 7) = 1−0, 99547−e
−2
2
7
7!
= 0, 0011
→ X ∼ B(n = 1800; p = 1/5000)

−∞
f(t)dt =









0 ; x < 30
x − 30
20
; x ∈ [30; 50]
1 ; x > 50
E(X) =
30 + 50
2
= 40; V (X) =
(50 − 30)
2
12
= 33, 333
P (X  45) = F (45) =
45 − 30
20
= 0, 75
X ∼ U(a, b)


3
b = 30 + 5
√
3
f(x) =





1
10
√
3
x ∈ (30 − 5
√
3; 30 + 5
√
3)
0 ; x /∈ (30 − 5
√
3; 30 + 5
√
3)
P (X  32) =
+∞

32
f(x)dx =
30+5

1 − e
−x/1500
; x  0
P (X < 1500) = F (1500) = 1 − e
−1500/1500
= 1 − e
−1
= 0, 632
f(x) =



5e
−5x
; x  0
0 ; x < 0
⇒ λ = 5
P (0, 4 < X < 1) =
1

0,4
f(x)dx =
1

0,4
5e
−5x
dx = −e
−5x


+∞

2
f(x)dx =
+∞

2
(1/3)e
−(1/3)x
dx = −e
−(1/3)x


+∞
2
= 0, 51342
P (−2, 33 < U < 2, 33) = P (U > −2, 33) − P (U > 2, 33) = 1 − α − α
= 1 − 2.0, 0099 = 0, 9802
α u
α
= 2, 33 ⇒ α = 0, 0099
P (−2 < U < 1) = P (U > −2)−P(U > 1) = 1−α
1
−α
2
= 1−0, 0228 −0, 1587
= 0, 8185
α
1
u

α
= 3, 02 ⇒ α = 0, 0013
P (U < 2, 5) = 1 − P (U > 2, 5) = 1 − 0, 0062 = 0, 9938
→ X ∼ N(µ, σ
2
)
E(X) = µ = 100(g); σ = 1(g)
P (98 < X < 102) = P

98 − 100
1
<
X −100
1
<
102 − 100
1

= P (−2 < U < 2)
= P (U > −2) − P (U > 2) = 1 − 2α
α P (U > 2) = α ⇒ u
α
= 2 ⇒ α = 0, 0228
⇒ P (98 < X < 102) = 1 − 2α = 1 − 2.0, 0228 = 0, 9544
P
b
= 1 − P
a
= 1 − 0, 9544 = 0, 0456
⇒ X ∼ N(µ = 15000, σ

, α
2
u
α
1
= 1 → α
1
= 0, 1587; u
α
2
= 2 → α
2
= 0, 0013
⇒ P (14500 < X < 16500) = 1 − 0, 1587 − 0, 0013 = 0, 84
→ X ∼ N(µ = 200; σ
2
= 40
2
)
P (X > 250) = P

X −200
40
>
250 − 200
40

= P (U > 1, 25) = 0, 1056
P (X < 180) = P (U <
180 − 200

18
σ
< U <
18
σ
) = 1 − 2P (U >
18
σ
)
⇔ P (U >
18
σ
) = 0 ⇔
18
σ
≈ 5 ⇔ σ = 3, 6
P (X > 55) = P (U >
55 − 50
3, 6
) = P (U > 1, 39) = 0, 0823
P (X < 40) = P (U <
40 − 50
3, 6
) = P (U < −2, 78) = P (U > 2, 78) = 0, 0027
X =
1
n

n
i=1

2
n
→ σ
X
=
σ
√
n
⇒
X ∼ N(m,
σ
2
n
)
P (|
X −m| < ε) = 2Φ
0
(
ε
σ
X
) = 2Φ
0
(
ε
√
n
σ
)
X ∼

(a, b) = (|X −µ| < 2σ) = (µ − 2σ < X < µ + 2σ)
⇒ a = µ − 2σ = 810 − 2.9 = 792; b = µ + 2σ = 810 + 2.9 = 828
f ∼ N(µ = p; σ
2
=
p(1 − p)
n
)
P (|f − p|  0, 02) = 0, 7698
P (|f − p|  0, 02) = 2Φ
0
(
0, 02
σ
) = 0, 7698
⇔ Φ
0
(
0, 02
σ
) = 0, 3849 ⇒
0, 02
σ
= 1, 2 ⇒ σ =
0, 02
12
= 0, 01667
n =
p(1 − p)
σ

< X < d
2
) = 1 − P

d
1
−
d
1
+ d
2
2
d
2
− d
1
4
< U <
d
2
−
d
1
+ d
2
2
d
2
− d
1

P (Z = 150) = P (X  a) = 1 − P (X < a); P (Z = −350) = P (X < a)
E(Z) = −350.P (X < a) + 150(1 − P (X < a)) = 150 − 500.P (X < a)
E(Z) = 50 ⇔ 150−500.P (X < a) = 50 ⇔ P(X < a) = 0, 2 ⇔ P (U <
a − 4, 2
1, 8
) = 0, 2
⇔ P (U >
4, 2 − a
1, 8
) = 0, 2 ⇒
4, 2 − a
1, 8
= 0, 84 ⇒ a = 4, 2 −18.0, 84 = 2, 688
X ∼ P (λ = 12)
P (X > 8) = 1 − P (X  8) = 1 −
8

i=0
e
−12
12
i
i!
= 0, 845
P (X > 15) = 1 − P (X  15) = 1 −
15

i=0
e
−12

11 − a
2
= 1, 28 ⇒ a = 8, 44
⇒ X ∼ N(µ; σ
2
= 9
2
)
0, 8413 = P (X  84) = P (U 
84 − µ
9
) ⇒ P (U >
84 − µ
9
) = 0, 1587 ⇒
84 − µ
9
= 1
⇒ µ = 75
P (X  80) = P (U >
80 − 75
9
) = P (U > 5/9) = 0, 2877 ⇒ P (X < 80) = 0, 7123
P () = 1 − P (X < 80)
3
= 1 − 0, 7123
3
= 0, 6386
→ X ∼ N(µ
1

= 1 + 0 = 1
V (X) = e(X
2
) − [E(X)]
2
= E(α + βZ + δZ
2
)
2
− (α + δ)
2
= α
2
+ β
2
E(Z
2
) + δ
2
E(Z
4
) + 2αβE(Z) + 2αδE(Z
2
) + 2βδE(Z
3
) − (α + δ)
2
= δ
2
E(Z

z
3
e
−z
2
/2
dz = 0
E(Z
4
) =
1
√
2π
+∞

−∞
z
4
e
−z
2
/2
dz
2
√
2π
+∞

0
z

/2)
V (X) = 3δ
2
+ β
2
− δ
2
= β
2
+ 2δ
2
Γ(a) =
+∞

0
x
a−1
e
−x
dx
Γ(a + 1) = aΓ(a); Γ(1/2) =
√
π
⇒ Γ(5/2) = (3/2)Γ(3/2) = (3/2).(1/2)Γ(1/2) = (3/4)
√
π
X ∼ N(µ = 8%; σ
2
= 10
2

P (X = 2) = C
2
20
.0, 4
2
.0, 6
18
= 0, 003087
P (X  2) = C
0
20
.0, 4
0
.0, 6
20
+ C
1
20
.0, 4
1
.0, 6
19
+ 0, 003087 = 0, 0036115
p ≈ f = 0, 1
p = 0, 4
→ X ∼ N(µ; σ
2
)
0, 2 = P (X > 20) = P (U >
20 − µ

2,12
= 4, 717
µ = 16, 038
P (X  14) = P (U 
14 − 16, 038
4, 717
= −0, 43) = 1−P (U > 0, 43) = 1−0, 3336 = 0, 6664
X ∼ N(µ
x
= 8; σ
2
x
= 0, 3
2
); Y ∼ N(µ
y
= 4; σ
2
y
= 0, 2
2
)
P
a
= P (|X −µ
x
|  0, 1; |Y −µ
y
|  0, 1) = P (|X −µ
x

3600 − 4300
250
) = P (U  −2, 8) = P (U > 2, 8) = 0, 0026
2.3600 = 7200
µ
1
Y ∼ N(µ
1
; 250
2
)
0, 0026 = P (Y  7200) = P (U 
7200 − µ
1
250
) = P (U >
µ
1
− 7200
250
)
⇒
µ
1
− 7200
250
= u
0,0026
= 2, 8 ⇒ µ
1

) + P (H
2
)P (B/H
2
)
• P (H
1
); P (H
2
)
→ Y ∼ B(n = 2; p = 0, 5)
P (H
2
) = P (Y = 0) = 0, 5
2
= 0, 25
P (H
1
) = 1 − P (H
2
) = 1 − 0, 25 = 0, 75
• P (B/H
1
)
→ Z ∼ B(n = 4; p = 0, 5)
P (B/H
1
) = P (Z  1) = 1 − P (Z = 0) = 1 − C
0
4

A
i
(i = 1, 2) A
i
= (|X|  2; |Y |  4)
P (A
i
) = P (|X|  2; |Y |  4) = P (|X|  2)P (|Y |  4) = 2Φ
0
(
2
4
)2Φ
0
(
4
6
)
= 2.0, 1915.2.0, 2486 = 0, 190428
⇒ P (
A
i
) = 1 − P (A
i
) = 1 − 0, 190428 = 0, 809572
→ B = A
1
A
2
+ A

72
.0, 083
1
.0, 917
71
= 0, 0127
P (X = 2) = C
2
72
.0, 083
2
.0, 917
70
= 0, 0409
E(X) = np = 72.0, 083 = 5, 976
X
A
, X
B
→ X
A
∼ N(µ
A
= 11; σ
2
A
= 4
2
); X
B

2
V (X
B
) = p
2
.4
2
+(1−p)
2
.2, 6
2
= 22, 76p
2
−13, 52p+6, 76
f(p) = 22, 76p
2
− 13, 52p + 6, 76
f
′
(p) = 45, 52p − 13, 52 = 0 ⇔ p =
13, 52
45, 52
≈ 0, 297
f”(p) = 45, 52 > 0