Томусяк А.А., Трохименко В.С. Теорія ймовірностей
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p
•
•
•
X, Y, Z x, y, z
X
ω x
x
X = x
X x
x
P (X = x)
x
1
, x
2
, . . . , x
n
X
p
1
= P (X = x
1
), p
2
= P (X = x
2
), . . . , p
n
= P (X = x
n
),
X
x
1
, x
2
, . . . , x
n
p
1
, p
2
, . . . , p
n
i
ω
1
ω
2
ω
3
ω
4
ω
5
ω
6
ω
7
ω
8
ω
9
ω
10
ω
11
ω
12
ω
13
ω
14
ω
15
ω
16
X
ω
16
ω
12
ω
13
ω
14
ω
15
ω
6
ω
7
ω
8
ω
9
ω
10
ω
11
ω
2
ω
3
ω
4
ω
5
ω
1
1
16
p
0
= P (X = 0) =
1
16
, p
1
= P (X = 1) =
4
16
, p
2
= P (X = 2) =
6
16
,
p
3
= P (X = 3) =
4
16
, p
4
= P (X = 4) =
1
16
.
X
P
1
16
4
16
6
16
4
16
1
16
¤
X x
1
x
2
. . . x
n
P p
1
p
2
. . . p
n
p
1
, p
2
, . . . , p
n
n
P
k=1
p
k
∞
P
k=1
p
k
X
a
P (X < a) =
X
k:x
k
<a
p
k
;
X
(a; b)
P (a < X < b) =
X
k:x
k
∈(a;b)
p
k
;
X
b
P (X > b) =
X
k:x
k
>b
p
k
.
X Y
(X, Y )
(x, y) X Y
X
x
1
, x
2
, . . . , x
m
Y y
1
, y
2
, . . . , y
n
(X, Y )
P (X = x
i
, Y = y
j
) = p
ij
,
i = 1, m j = 1, n
(X, Y )
X\Y y
1
y
2
. . . y
n
x
1
p
11
p
12
. . . p
1n
x
2
p
21
p
22
. . . p
2n
x
m
p
m1
p
m2
. . . p
mn
(X, Y )
P (X = x
i
) =
n
X
j=1
P (X = x
i
, Y = y
j
) =
n
X
j=1
p
ij
,
P (Y = y
j
) =
m
X
i=1
P (X = x
i
, Y = y
j
) =
m
X
i=1
p
ij
.
X, Y
X Y
P (X = 2) =
1
36
, P (Y = 0) =
1
6
.
(X, Y )
X Y
X
Y
P (X = x
i
, Y = y
j
) = P (X = x
i
)P (Y = y
j
).
X Y
X x
1
x
2
. . . x
m
P p
(1)
1
p
(1)
2
. . . p
(1)
m
,
Y y
1
y
2
. . . y
n
P p
(2)
1
p
(2)
2
. . . p
(2)
n
(x
i
, y
j
)
p
ij
= P (X = x
i
, Y = y
j
) = P (X = x
i
)P (Y = y
j
) = p
(1)
i
p
(2)
j
.
X
1
, X
2
, . . . , X
n
n
k
n (X
1
, X
2
, . . . , X
n
)
(x
i
1
, x
i
2
, . . . , x
i
n
)
P (X
1
= x
i
1
, X
2
= x
i
2
, . . . , X
n
= x
i
n
) =
= P (X
1
= x
i
1
)P (X
2
= x
i
2
) ···P (X
n
= x
i
n
).
X + Y
1
2
(X
2
+ Y
2
) X Y
X
P
,
Y −1
P
(X, Y )
X\Y −1
.
X+Y
−1
P (X + Y = −1) = P (X = 0, Y = −1) = 0, 15;
P (X + Y = 0) = P (X = 0, Y = 0 X = 1, Y = 0) =
= 0, 15 + 0, 20 = 0, 35;
P (X + Y = 1) = P (X = 0, Y = 1 X = 1, Y = 0) =
= 0, 15 + 0, 20 = 0, 35;
P (X + Y = 2) = P (X = 1, Y = 1) = 0, 15.
X + Y −1
P
X + Y
1
2
(X
2
+ Y
2
)
1
2
P
³
1
2
(X
2
+ Y
2
) = 0
´
= P (X = 0, Y = 0) = 0, 20
P
³
1
2
(X
2
+ Y
2
) =
1
2
´
=
= P (X = 0, Y = −1, X = 0, Y = 1, X = 1, Y = 0) =
= 0, 15 + 0, 15 + 0, 20 = 0, 5;
P
³
1
2
(X
2
+ Y
2
) = 1
´
= P (X = 1, Y = −1 X = 1, Y = 1) =
= 0, 15 + 0, 15 = 0, 30.
1
2
³
X
2
+ Y
2
´
0
P
. ¤