Задания
.pdfn
n
n m
1; 2; . . . ; n
3n
n
n n
1 2
n + . . . + n =n n
k 1 k
X; Y
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0 < u < 1 0 < v < 1 |
P {X < u, Y < v} = P {X < u}P {Y < v} = uv; |
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0 < t < 1 |
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1) P {|X − Y | < t}; |
2) P {XY < t}; |
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3) P {max(X, Y ) < t}; |
4) P {min(X, Y ) < t}; |
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P {X + Y < t} |
0 < t < 2 |
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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 |
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i |
A |
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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}
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1 y |
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− |
y |
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2) −| |
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f(y) = cos y f(y) |
1 |
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f y e= |
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f(y) = e |
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{
f(y) =
Cy2, y [0, 1],
0, y ̸ [0, 1].
C
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λ = n |
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α,λ |
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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], |
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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 } |
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X Y |
P{X = 0} = P{X = 1} = 1/2 |
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1 |
X |
2 |
= XY− |
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= 2X + 1 |
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X |
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2 = XY− [X] |
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1 = [X] |
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4 |
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ln |
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√ |
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Y |
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= α− |
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5 = |
YX |
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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
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∫ |
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X2 |
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∫ |
∞ |
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4dt |
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X |
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−2 |
t |
≥ |
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E |
X 1 |
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∞ |
3 |
dt = 3/2 |
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3 |
t |
= 1 |
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X1 |
= 1 |
− (3 |
/ |
2) |
2 |
< |
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− |
1 |
3t− |
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E− |
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D− |
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ρ (X, Y )
2) X ρ (X, X
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T |
Z |
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, Z2) |
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Z = (Z1 |
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T |
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X = (X1, X2) |
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8 |
18 |
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45 |
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18 |
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2 + X2 C X
1 2
CY,X ) (X
1 2
Y
1, X2, ... X
λ
X12 + ... + Xn2 |
( |
X1 + ... + Xn |
)2 |
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? |
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n |
n |
X
DX
n
−1
exp(2it − 2t2)
n n → ∞
1, X2, . . . X
n → ∞
X12 + . . . + Xn2
n ;
√
X12 + . . . + Xn2
n ;
1 ( |
1 |
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+ . . . + |
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; |
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n |
1 +( |
1 |
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1 + |
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n ) |
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2 |
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+ . . . + Xn) . |
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arctg (X1 |
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n |
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−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. |
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0 |
f(x) = |
{ 3x2 e−x3/θ |
x > 0; |
θ |
x ≤ 0. |
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0 |
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{ 1 |
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√ |
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2θ√ |
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e− x/θ |
x > 0; |
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f(x) = |
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x |
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0 |
x ≤ 0. |
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P{X = 1} = p2, P{X = 2} = 2p(1 − p), P{X = 3} = (1 − p)2.
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{ 3√ |
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√ |
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x |
x > 0; |
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f(x) = |
2θ |
e−x x/θ |
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0 |
x ≤ 0. |
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f(x) = |
{ 1 x−(θ+1)/θ |
x > 1; |
θ |
x ≤ 1. |
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0 |
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{ |
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1)x−θ |
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f(x) = |
(θ |
− |
x > 1; |
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0 |
x ≤ 1. |
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{ |
x |
e−x/θ |
x > 0; |
f(x) = |
2 |
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θ |
x ≤ 0. |
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0 |
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{ |
x2 |
e−x/θ |
x > 0; |
f(x) = |
3 |
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2θ |
x ≤ 0. |
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0 |
P{X = 0} = (1 − p)3, P{X = 1} = 3p(1 − p)2, P{X = 2} = 3p2(1 − p), P{X = 3} = p3.
{ √ |
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2 |
e−x2/θ |
x > 0; |
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f(x) = πθ |
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0 |
x ≤ 0. |
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f(x) = |
{ 4x3 e−x4/θ |
x > 0; |
θ |
x ≤ 0. |
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0 |
f(x) = |
{ 2 e−2x/θ |
x > 0; |
θ |
x ≤ 0. |
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0 |