Томусяк А.А., Трохименко В.С. Теорія ймовірностей
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F (x)
f(x) F (x)
F
0
(x) = f(x).
f(x)
(X, Y )
(x, y)
X
x Y
y
(x, y)
xOy
X x
Y y
X Y
F (x, y) = P (X < x, Y < y).
(X, Y )
(x, y)
Π = {(x, y) |a 6 x < b, c 6 y < d}
M
1
(a, c) M
2
(b, c) M
3
(b, d) M
4
(a, d)
M
2
M
3
M
3
M
4
X < b Y < d (X, Y )
(b, d)
(X, Y ) ∈ Π Π
X < a Y < d (X, Y )
(a, d) X < b Y < c (X, Y )
(b, c)
P (X < c, Y < d) =
= P ((X, Y ) ∈ Π) + P (X < a, Y < d X < b, Y < c) =
= P ((X, Y ) ∈ Π) + P (X < a, Y < d) + P (X < b, Y < c) −
−P (X < a, Y < b).
P ((X, Y ) ∈ Π) = P (a 6 X < b, c 6 Y < d) =
= F (b, d) − F (a, d) − F (b, c) + F (a, b).
1
◦
x y
0 6 F (x, y) 6 1.
2
◦
F (x, y)
F (x
1
, y) 6 F (x
2
, y), x
1
< x
2
,
F (x, y
1
) 6 F (x, y
2
), y
1
< y
2
.
3
◦
lim
x→−∞
F (x, y) = lim
y→−∞
F (x, y) = lim
x → −∞
y → −∞
F (x, y) = 0
lim
x → +∞
y → +∞
F (x, y) = 1.
4
◦
lim
y→+∞
F (x, y) = F
1
(x), lim
x→+∞
F (x, y) = F
2
(y),
F
1
(x), F
2
(y)
X Y
(X, Y ) X Y
(X, Y )
X 6 0 Y 6 0
0 < x 6 1
0 < y 6 1 X < x Y < y
{(u, v)|0 6 u < x, 0 6 v < y},
xy x > 1 0 < y 6 1
X < x Y < y
{(u, v)|0 6 u 6 1, 0 6 v < y}
y 0 < x 6 1 y > 1 X < x
Y < y
{(u, v)|0 6 u < x, 0 6 v 6 1} x
x > 1 y > 1 X < x Y < y
F (x, y) =
0, x 6 0 y 6 0,
xy, 0 < x 6 1 0 < y 6 1,
y, x > 1 0 < y 6 1,
x, 0 < x 6 1 y > 1,
1, x > 1 y > 1. ¤
(X, Y )
F (x, y) (x
0
, y
0
)
x
0
∆x > 0 y
0
∆y > 0
P (x
0
6 X < x
0
+ ∆x, y
0
6 Y < y
0
+ ∆y) =
= F (x
0
+ ∆x, y
0
+ ∆y) − F (x
0
, y
0
+ ∆y) − F (x
0
+ ∆x, y
0
) +
+F (x
0
, y
0
).
P (x
0
6 X < x
0
+ ∆x, y
0
6 Y < y
0
+ ∆y)
∆x∆y
,
lim
∆x → 0
∆y → 0
P (x
0
6 X < x
0
+ ∆x, y
0
6 Y < y
0
+ ∆y)
∆x∆y
,
(x
0
, y
0
)
f(x
0
, y
0
)
f(x
0
, y
0
) =
= lim
∆y→0
1
∆y
µ
lim
∆x→0
F (x
0
+ ∆x, y
0
+ ∆y) − F (x
0
, y
0
+ ∆y)
∆x
−
− lim
∆x→0
F (x
0
+ ∆x, y
0
) − F (x
0
, y
0
)
∆x
¶
=
= lim
∆y→0
F
0
x
(x
0
, y
0
+ ∆y) − F
0
x
(x
0
, y
0
)
∆y
= F
00
xy
(x
0
, y
0
),
F (x, y)
(x
0
, y
0
)
f(x
0
, y
0
) = F
00
xy
(x
0
, y
0
).
f(x, y)
+∞
Z
−∞
+∞
Z
−∞
f(x, y)dx dy = 1,
f(x, y)
F (x, y) =
x
Z
−∞
y
Z
−∞
f(u, v)du dv,
f(x, y) D
(X, Y )
D
P ((X, Y ) ∈ D) =
ZZ
D
f(x, y)dx dy.
f(x, y)
F
1
(x) =
x
Z
−∞
+∞
Z
−∞
f(u, v)du dv, F
2
(y) =
+∞
Z
−∞
y
Z
−∞
f(u, v)du dv
X Y
f
1
(x) =
+∞
Z
−∞
f(x, v)dv, f
2
(y) =
+∞
Z
−∞
f(u, y)du
X Y
F (x, y) = F
1
(x)F
2
(y).
F
1
(x) =
(
0, x 6 0,
x, 0 < x 6 1,
1, x > 1,
F
2
(y) =
(
0, y 6 0,
y, 0 < y 6 1,
1, y > 1.
(X, Y )
(X, Y )
X
Y
f(x, y) = f
1
(x)f
2
(y).
X
f(x)
X
M(X) =
+∞
Z
−∞
xf(x)dx
X
D(X) =
+∞
Z
−∞
(x − M (X))
2
f(x)dx.
X
f(x) c 6= 0
cX f(x)
M(cX) =
+∞
Z
−∞
cxf(x)dx = c
+∞
Z
−∞
xf(x)dx = cM(X).
1
◦
2
◦
(X, Y ) f(x, y)
X Y
M(X) =
+∞
Z
−∞
xf
1
(x)dx =
+∞
Z
−∞
x
+∞
Z
−∞
f(x, v)dv dx,
M(Y ) =
+∞
Z
−∞
yf
2
(x)dy =
+∞
Z
−∞
y
+∞
Z
−∞
f(u, y)du dy.
Z
F (z) = P (Z < z) = P (X + Y < z) =
ZZ
u+v<z
f(u, v)du dv =
=
+∞
Z
−∞
du
z−u
Z
−∞
f(u, v)dv =
+∞
Z
−∞
du
z
Z
−∞
f(u, t −u)dt =
=
z
Z
−∞
+∞
Z
−∞
f(u, t − u)du dt.
v = t − u
z−u
Z
−∞
f(u, v)dv
Z
f(z) =
+∞
Z
−∞
f(u, z − u)du.
M(Z) =
+∞
Z
−∞
zf (z)dz =
+∞
Z
−∞
+∞
Z
−∞
(z − u + u)f(u, z − u)du dz =
=
+∞
Z
−∞
+∞
Z
−∞
(z − u)f(u, z − u)du dz +
+∞
Z
−∞
+∞
Z
−∞
uf(u, z − u)du dz =
=
+∞
Z
−∞
+∞
Z
−∞
tf(u, t)du dt +
+∞
Z
−∞
+∞
Z
−∞
uf(u, t)du dt =
=
+∞
Z
−∞
t
+∞
Z
−∞
f(u, t)du dt +
+∞
Z
−∞
+∞
Z
−∞
f(u, t)dt du =
=
+∞
Z
−∞
tf
2
(t)dt +
+∞
Z
−∞
uf
1
(u)du = M(Y ) + M(X). ¤
3
◦
X Y
M(XY ) = M (X)M(Y ).
4
◦
D(cX) = c
2
D(X).
5
◦
D(X) = M( X
2
) − (M (X))
2
.