Томусяк А.А., Трохименко В.С. Теорія ймовірностей
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x
∞
X
n=0
x
n
n!
= e
x
,
λ
M(X) = D(X) = λ.
n
P (X = n) =
2
n
n!
e
−2
.
P (X = 3) =
2
3
3!
e
−2
≈ 0, 1804,
P (X > 4) = 1−P (X < 4) = 1−
3
X
n=0
2
n
n!
e
−2
≈ 0, 0527. ¤
X
X
P x
p
3
= P (X = 3)
n
p 0 < p < 1 np
npq
p 0 < p < 1
λ > 0
X
M(X) = 2 , 3; M(X
2
) = 5, 9.
Y = −2X − 1
X
n = 80 D(X) = 15
M(X)
X > 20
•
•
•
•
X
x x
(−∞; x)
x
X x
F (x) = P (X < x).
x
1
x
2
x
1
< x
2
X < x
1
X
x
1
x
1
6 X < x
2
X
x
1
x
2
X < x
2
X
x
2
P (X < x
1
) + P (x
1
6 X < x
2
) = P (X < x
2
).
P (x
1
6 X < x
2
) = F (x
2
) − F (x
1
),
X
x
1
x
2
[x
1
; x
2
)
P (x
1
6 X < x
2
) = P (x
1
< X < x
2
) = P (x
1
< X 6 x
2
) =
= P (x
1
6 X 6 x
2
) = F (x
2
) − F (x
1
).
1
◦
F (x) x
1
x
2
x
1
< x
2
F (x
1
) 6 F (x
2
)
2
◦
F (x)
lim
x→−∞
F (x) = 0, lim
x→+∞
F (x) = 1.
3
◦
F (x)
x
0
∈ R
lim
x→x
0
−0
F (x) = F (x
0
).
1
◦
3
◦
1
◦
3
◦
F (x) =
1
π
arctg x +
1
2
arctg x
F (x)
lim
x→−∞
F (x) =
1
π
lim
x→−∞
arctg x +
1
2
=
1
π
³
−
π
2
´
+
1
2
= 0,
lim
x→+∞
F (x) =
1
π
lim
x→+∞
arctg x +
1
2
=
1
π
·
π
2
+
1
2
= 1.
1
◦
3
◦
X
P (X < x) =
1
π
arctg x +
1
2
.
F (x) =
1
π
arctg x +
1
2
X
X
h
−
π
4
;
π
4
i
P
³
−
π
4
6 X 6
π
4
´
= F
³
π
4
´
− F
³
−
π
4
´
=
=
1
π
arctg
π
4
+
1
2
−
µ
1
π
arctg
³
−
π
4
´
+
1
2
¶
=
2
π
. ¤
X
F (x) x
0
x
0
∆x > 0 F (x)
(x
0
; x
0
+ ∆x)
lim
∆x→0
∆F (x
0
)
∆x
= lim
∆x→0
P (x
0
< X < x
0
+ ∆x)
∆x
,
x
0
F (x) x
0
lim
∆x→0
∆F (x
0
)
∆x
= F
0
(x
0
),
P (x
0
< X < x
0
+ ∆x) ≈ F
0
(x
0
)∆x.
f(x)
+∞
Z
−∞
f(x)dx = 1,
f(x)
F (x) =
x
Z
−∞
f(t)dt.
F (x) x ∈ R
x
1
< x
2
F (x
2
) − F (x
1
) =
x
2
Z
−∞
f(x)dx −
x
1
Z
−∞
f(x)dx =
x
2
Z
x
1
f(x)dx > 0,
f(x) F (x)
lim
x→−∞
F (x) = lim
x→−∞
x
Z
−∞
f(t)dt = 0,
lim
x→+∞
F (x) = lim
x→+∞
x
Z
−∞
f(t)dt =
+∞
Z
−∞
f(x)dx = 1.
F (x)
f(x)
F (x)
1
◦
3
◦
f(x)
+∞
Z
−∞
f(x)dx = 1,