Дронов С.В. Математические методы в психологии
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σ
2
ρ
χ
2
n × m n m
[A]
n,m
n = m
n
A
A
t
[A
t
]
m,n
i, j a
t
i,j
= a
j,i
A A = A
t
n = m a
i,i
n = 1 [A]
1,n
m = 1 [A]
m,1
m, n
~a
~a
t
A n | A |
n = 2
a
1,1
a
1,2
a
1,1
a
2,2
= a
1,1
a
2,2
− a
1,2
a
2,1
∆
n+1
n + 1
∆
n+1
=
n+1
X
j=1
(−1)
j+1
a
1,j
∆
1,j
n
,
∆
1,j
n
n ∆
n+1
j
[A]
n,m
, [B]
m,l
, C = AB. [C]
n,l
c
i,j
=
m
X
k=1
a
i,k
b
k,j
, i = 1, ..., n, j = 1, ...l.
~a n A~a m
A, B
| AB | = | A | | B |, | A
t
| = | A | .
[I]
n,n
n
[A]
n,n
AI = IA = A
A
−1
A
AA
−1
= A
−1
A = I.
A ~a λ
A~a = λ~a ~a
λ
| A − λI | = 0.
α
A(α~a) = α(A~a) = λ(α~a),
α~a
~a
k~ak =
v
u
u
t
n
X
j=1
a
2
j
−
~a
n n
n × n n
n × m m
n
A A
−1
= A
t
n A
~a,
~
b
ϕ = arccos
< ~a,
~
b >
k~akk
~
bk
,
< ~a,
~
b > =
P
k
a
k
b
k
~a,
~
b < ~a,
~
b > = 0.
< ~a,
~
b > = ~a
t
~
b,
C = ~a
~
b
t
n × n c
i,j
=
a
i
b
j
, i, j = 1, ..., n
X
X x
1
x
n
p
1
p
n
x
1
, ..., x
n
p
1
, ..., p
n
F (x) = P (X < t) =
X
j:x
j
<t
p
j
.
MX =
n
X
k=1
x
k
p
k
.
DX = M(X −MX)
2
.
M(X + Y ) = MX + MY, M(αX) = αMX, D(αX) = α
2
DX,
X, Y
MXY = MX MY, D(X + Y ) = DX + DY.
σ =
√
DX X
(MX − 3σ, MX + 3σ)
X
ρ(X, Y ) =
cov(X, Y )
√
DXDY
,
cov(X, Y ) = MXY − MXMY = M(X −MX)(Y − MY )
X Y
ρ
X Y ρ(X, Y ) = 0
ρ
| ρ(X, Y ) | < 1/3
X Y 1/3 ≤| ρ(X, Y ) | < 2/3 | ρ(X, Y ) | ≥ 2/3
ρ > 0
a σ
2
p(x) =
1
σ
√
2π
exp
−
(x − a)
2
2σ
2
.
a = 0, σ = 1
ϕ(x) =
1
√
2π
exp(−x
2
/2).
Φ(t) =
t
Z
−∞
ϕ(x) dx
X a σ
2
Y =
(X − a)/σ
P (X < t) = P
X − a
σ
<
t − a
σ
= Φ
t − a
σ
.
n
~x x
1
, ..., x
n
M~x
~x
C
c
i,j
= cov(x
i
, x
j
), i, j = 1, ..., n.
c
j,j
= Dx
j
C = M(~x − M~x)(~x − M~x)
t
.
n n
x
1
x
2
x
3
, x
4
, x
5
x
1
, x
2
x
3
, x
4
, x
5
x
3
x
4
, x
5
x
1
x
2
x
3
x
4
x
5
X n
¯
X =
1
n
n
X
i=1
x
i
S
2
=
1
n
n
X
i=1
(x
i
−
¯
X)
2
=
1
n
n
X
i=1
x
2
i
−
1
n
n
X
i=1
x
i
!
2
−
x
¯
X
S
2
Mx, Dx
S
2
n
S
2
0
=
1
n − 1
n
X
i=1
(x
i
−
¯
X)
2
.
AsX =
1
nS
3
n
X
i=1
(x
i
−
¯
X)
3
, ExX =
1
nS
4
n
X
i=1
(x
i
−
¯
X)
4
− 3.
S S
2
X
(1)
= min{x
1
, ..., x
n
}, X
(n)
= max{x
1
, ..., x
n
}
T = X
(n)
− X
(1)
.