Задания
.pdfn
n
n m
1; 2; . . . ; n
3n
n
n n
1 2
n + . . . + n =n n
k 1 k
X; Y
|
|
0 < u < 1 0 < v < 1 |
P {X < u, Y < v} = P {X < u}P {Y < v} = uv; |
||
0 < t < 1 |
|
|
1) P {|X − Y | < t}; |
2) P {XY < t}; |
|
3) P {max(X, Y ) < t}; |
4) P {min(X, Y ) < t}; |
|
P {X + Y < t} |
0 < t < 2 |
|
|
|
l |
X, Y Z
2n 2n
n
l
2l/3
1/2
1/2
A
A
C
R n
n
p |
m + l |
l |
1; . . . ; An A |
i = P ( A i ) |
i = 1; . . . ; n |
|
|
|
|
|
|
i |
A |
|
i A
i A
A B
B
A
k
l
A
C
n
m/n
k 1 − q
n
BBBB CCCC
ABCA
λk e−λ k
k!
λ
m
P{a < X < b} P{a ≤ X < b} P{a < X ≤ b} P{a ≤ X ≤ b}
( |
1 y |
| |
− |
y |
|
≡ |
|
2) −| |
|
f(y) = cos y f(y) |
1 |
||||
f y e= |
|
f(y) = e |
|
||||
{
f(y) =
Cy2, y [0, 1],
0, y ̸ [0, 1].
C
|
λ = n |
|
α,λ |
|
|
|
|
|
l
R
X \ Y
X Y
2 Z = Y
n
1
n Y k k Y k =
X
X
f(t) = |
{ θtθ−1, t [0, 1], |
|
|
0, |
t ̸ [0, 1]. |
X Y
X
X
= X2 = sinY X
1 2
max(0, X)
kP}{=XP={Yy = yk} = pk k ≥ 1 |
X Y |
P{X = Y } |
|
X Y |
P{X = 0} = P{X = 1} = 1/2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
X |
2 |
= XY− |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
= 2X + 1 |
1 |
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
X |
|
2 = XY− [X] |
|
|
|
|
|
|
|
|
|
|
|
|
1 = [X] |
|
|
Y |
|||||
|
3 |
= |
|
2 |
4 |
|
1 |
ln |
|
√ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||
Y |
X |
|
= α− |
X |
5 = |
YX |
|
|
|
|
|
|
||||||
|
|
|
|
Y |
|
|
|
|
|
|
|
|
|
|||||
X
Y
X Y
P{X = 2Y }
X Y
m
∑ |
∑ |
∞ |
∞ |
P (X ≥ k) EX = |
P ( X = k ) = 1 |
k=1 |
k =1 |
X Y X
Y
2X + 3Y
[a; b]
Y
1 n
2013 EX X
k
X
Ck−10, k = 1, 2, . . . C
X
ρ (X, X + Y )
X
β
Y = eβX
|
∫ |
|
|
|
X2 |
|
∫ |
∞ |
|
|
4dt |
|
|
X |
|
|
|
|
|
|
−2 |
t |
≥ |
1 |
||
E |
X 1 |
= |
∞ |
3 |
dt = 3/2 |
|
= |
|
3 |
t |
= 1 |
|
X1 |
= 1 |
− (3 |
/ |
2) |
2 |
< |
0 |
|
|
||||
− |
1 |
3t− |
|
E− |
|
1 |
− |
|
|
D− |
|
|
|
|
|
|
||||||||||
ρ (X, Y )
2) X ρ (X, X
|
|
|
|
|
|
|
T |
Z |
|
|
|
||
( |
|
) |
|
|||
|
, Z2) |
|
|
|||
Z = (Z1 |
|
|
||||
|
|
T |
|
|
||
X = (X1, X2) |
|
|
||||
|
|
|
|
|
||
|
|
|
8 |
18 |
|
|
|
|
|
45 |
|||
|
|
Z |
18 |
|
||
2 + X2 C X
1 2
CY,X ) (X
1 2
Y
1, X2, ... X
λ
X12 + ... + Xn2 |
( |
X1 + ... + Xn |
)2 |
|
− |
? |
|||
n |
n |
X
DX
n
−1
exp(2it − 2t2)
n n → ∞
1, X2, . . . X
n → ∞
X12 + . . . + Xn2
n ;
√
X12 + . . . + Xn2
n ;
1 ( |
1 |
|
|
|
1 |
) |
||
|
|
|
+ . . . + |
|
|
|
; |
|
X |
|
|
X |
|
||||
n |
1 +( |
1 |
|
|
1 + |
|
n ) |
|
|
2 |
|
+ . . . + Xn) . |
|||||
arctg (X1 |
||||||||
|
n |
|
|
|
|
|
||
−8
1, . . . , xn) x = (x
P{X = 0} = (1 − p)2, P{X = 1} = 2p(1 − p), P{X = 2} = p2.
f(x) = |
{ 2x e−x2/θ |
x > 0; |
θ |
x ≤ 0. |
|
|
0 |
f(x) = |
{ 3x2 e−x3/θ |
x > 0; |
θ |
x ≤ 0. |
|
|
0 |
|
{ 1 |
|
√ |
|
|
|
|
|
|
|
|
||||
|
|
2θ√ |
|
e− x/θ |
x > 0; |
||
f(x) = |
|
x |
|||||
|
|
|
0 |
x ≤ 0. |
|||
|
|
|
|
||||
P{X = 1} = p2, P{X = 2} = 2p(1 − p), P{X = 3} = (1 − p)2.
|
{ 3√ |
|
|
√ |
|
|
|
|
|
x |
x > 0; |
||||||
|
|
|
||||||
f(x) = |
2θ |
e−x x/θ |
||||||
0 |
x ≤ 0. |
|||||||
|
|
|
|
|||||
f(x) = |
{ 1 x−(θ+1)/θ |
x > 1; |
θ |
x ≤ 1. |
|
|
0 |
|
{ |
|
1)x−θ |
|
f(x) = |
(θ |
− |
x > 1; |
|
|
0 |
x ≤ 1. |
||
|
|
|
|
{ |
x |
e−x/θ |
x > 0; |
f(x) = |
2 |
|||
|
θ |
x ≤ 0. |
||
|
0 |
|||
|
{ |
x2 |
e−x/θ |
x > 0; |
f(x) = |
3 |
|||
2θ |
x ≤ 0. |
|||
|
0 |
|||
P{X = 0} = (1 − p)3, P{X = 1} = 3p(1 − p)2, P{X = 2} = 3p2(1 − p), P{X = 3} = p3.
{ √ |
|
|
||
|
|
|
|
|
|
2 |
e−x2/θ |
x > 0; |
|
f(x) = πθ |
||||
0 |
x ≤ 0. |
|||
|
|
|||
f(x) = |
{ 4x3 e−x4/θ |
x > 0; |
θ |
x ≤ 0. |
|
|
0 |
f(x) = |
{ 2 e−2x/θ |
x > 0; |
θ |
x ≤ 0. |
|
|
0 |