Томусяк А.А., Трохименко В.С. Теорія ймовірностей
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X
X
M(X) =
n
X
k=1
x
k
p
k
.
∞
X
n=1
x
k
p
k
N X
x
1
N
1
x
2
N
2
x
n
N
n
x =
1
N
(x
1
N
1
+ x
2
N
2
+ ··· + x
n
N
n
) =
= x
1
N
1
N
+ x
2
N
2
N
+ ··· + x
n
N
n
N
.
N
N
1
N
N
2
N
N
n
N
p
1
, p
2
, . . . , p
n
x
X
X
D(X) = M((X − M (X))
2
).
D(X) =
n
X
k=1
(x
k
− M (X))
2
p
k
.
X
X
P
1
16
4
16
6
16
4
16
1
16
M(X) = 0 ·
1
16
+ 1 ·
4
16
+ 2 ·
6
16
+ 3 ·
4
16
+ 4 ·
1
16
=
=
1
16
(4 + 12 + 12 + 4) = 2,
D(X) = (0 − 2)
2
·
1
16
+ (1 − 2)
2
·
4
16
+ (2 − 2)
2
·
6
16
+
+(3 − 2)
2
·
4
16
+ (4 − 2)
2
·
1
16
=
= 4 ·
1
16
+ 1 ·
4
16
+ 0 ·
6
16
+ 1 ·
4
16
+ 4 ·
1
16
= 1.
1
◦
M(c) = c.
2
◦
M(cX) = cM(X).
3
◦
M(X + Y ) = M(X) + M (Y ).
(X, Y )
X
X x
1
x
2
. . . x
m
P
n
P
j=1
p
1j
n
P
j=1
p
2j
. . .
n
P
j=1
p
mj
,
Y
Y y
1
y
2
. . . y
n
P
m
P
i=1
p
i1
m
P
i=1
p
i2
. . .
m
P
i=1
p
in
,
M(X) =
m
X
i=1
x
i
P (X = x
i
) =
m
X
i=1
n
X
j=1
x
i
p
ij
,
M(Y ) =
n
X
j=1
y
j
P (Y = y
j
) =
n
X
j=1
m
X
i=1
y
j
p
ij
.
Z = X + Y z
1
, z
2
, . . . , z
r
k = 1, r
P (Z = z
k
) =
X
(i,j);x
i
+y
j
=z
k
p
ij
.
M(X + Y ) = M(Z) =
r
X
k=1
z
k
P (Z = z
k
) =
m
X
i=1
n
X
j=1
(x
i
+ y
j
)p
ij
=
=
m
X
i=1
x
i
n
X
j=1
p
ij
+
n
X
j=1
y
j
m
X
i=1
p
ij
= M(X) + M(Y ). ¤
4
◦
X Y
M(XY ) = M (X)M(Y ).
X
Y
p
ij
= P (X = x
i
, Y = y
j
) = P (X = x
i
)P (Y = y
j
).
M(XY ) =
m
X
i=1
n
X
j=1
x
i
y
j
p
ij
=
=
m
X
i=1
n
X
j=1
x
i
y
j
P (X = x
i
)P (Y = y
j
) =
=
m
X
i=1
x
i
P (X = x
i
)
n
X
j=1
P (Y = y
j
) = M(X)M(Y ). ¤
5
◦
6
◦
D(cX) = c
2
D(X).
7
◦
D(X) = M( X
2
) − (M (X))
2
.
D(X) = M((X − M (X))
2
) = M(X
2
− 2M (X)X + (M(X))
2
) =
= M(X
2
) − 2M (X)M(X) + (M (X))
2
= M(X
2
) − (M (X))
2
. ¤
8
◦
D(X + Y ) = D(X) + D(Y ).
M(X) = a, M(Y ) = b
3
◦
M(X +Y ) = a+b 7
◦
4
◦
D(X + Y ) = M ((X + Y )
2
) − (a + b)
2
=
= M(X
2
+ 2XY + Y
2
) − a
2
− 2ab − b
2
=
= M(X
2
) − a
2
+ M (Y
2
) − b
2
+ 2M (X)M (Y ) − 2ab =
= D(X) + D(Y ). ¤
3
◦
4
◦
8
◦
M
Ã
n
X
i=1
X
i
!
=
n
X
i=1
M(X
i
)
X
1
, X
2
, . . . , X
n
M
Ã
n
Y
i=1
X
i
!
=
n
Y
i=1
M(X
i
),
D
Ã
n
X
i=1
X
i
!
=
n
X
i=1
D(X
i
)
M(X) X
σ(X) =
p
D(X).
X
x
1
, x
2
, . . . , x
n
P (X = x
k
) =
1
n
.
X
P
1
6
1
6
1
6
1
6
1
6
1
6
n p 0 < p < 1
n
P (X = k) = C
k
n
p
k
q
n−k
,
q = 1 − p
n
A p A
n
M(X) =
n
X
k=0
kP (X = k),
D(X) =
n
X
k=0
(k − M(X))
2
P (X = k).
A n
A
X
n µ
1
, µ
2
, . . . , µ
n
µ
k
P q p
.
M(µ
k
) = 0 ·q + 1 · p = p,
D(µ
k
) = (0 − p)
2
q + (1 − p)
2
p = p
2
q + q
2
p = pq.
M(X) =
n
X
k=1
M(µ
k
) =
n
X
k=1
p = np,
D(X) =
n
X
k=1
D(µ
k
) =
n
X
k=1
pq = npq,
σ(X) =
√
npq.
N, M, n
0, 1, 2, . . . , min(M, n)
P (X = k) =
C
k
M
C
n−k
N−M
C
n
N
.
n
N M
M(X) = n ·
M
N
,
D(X) = n ·
M(N − M)(N − n)
N
2
(N − 1)
.
n N
N − n
N − 1
n
A
M
N
p 0 < p < 1
0, 1, 2, . . .
P (X = n) = pq
n
,
q = 1 − p
A A
p
∞
X
n=0
x
n
(−1; 1)
∞
X
n=0
x
n
=
1
1 − x
.
∞
X
n=1
nx
n−1
1
(1 − x)
2
M(X) =
∞
X
n=0
nP (X = n) = p
∞
X
n=1
nq
n
=
=
pq
(1 − q)
2
=
q
p
,
D(X) =
∞
X
n=0
³
n −
q
p
´
2
P (X = n) =
q
p
2
.
λ > 0
0, 1, 2, . . .
P (X = n) =
λ
n
n!
e
−λ
.
λ