Apuntes de Estadística: Fórmulas, Demostraciones y Ejemplos Resueltos

X̄
G.O
H.O
S2
S
Sp2
M.S
Sx̄
µ
ω2
ω
m
n
m→
n = 24 ↓
n = 37 ↓
↑
↑
↑
n
24 ↔ 4.89 ↓ m = 5
37 ↔ 6.08 ↓ m = 7
R
↗
R=
AL
L
As =
ALi + Li→1
,
2
s
=
Li + ALi+1
2
yi
yi =
ALi + Li
As + s
=
2
2
R
c = s ↗ As = m
c
fi
fr = fni
fr
x
y
n = 27
R = 115 ↗ 90 = 25
↑
m → 27 ↔ 5.2 ↓ m = 6
R
c= m
= 25
6 ↔ 4.16 ↓ c = 5
AL ↗ L
As ↗ s
fi
yi
X̄
n
X̄ =
x1 + x2 + · · · + xn
1!
=
xi
n
n
i=1
m
f 1 y1 + f 2 y 2 + · · · + f m y m
1!
X̄ =
=
f i yi
n
n
i=1
wi
"k
X̄ = "i=1
k
X̄ =
X̄ =
w i xi
i=1 wi
x̄ m
ȳ n
mx̄ + nȳ
m+n
(5 ↘ 60) + (3 ↘ 70) + (3 ↘ 65) + (4 ↘ 60) + (2 ↘ 80)
300 + 210 + 195 + 240 + 160
1105
=
=
= 65
5+3+3+4+2
17
17
n
n
= x( n+1 )
2
=
x( n ) +x( n +1)
2
2
2
n
2
= Lm +
#n
2 ↗ nm
fm
$
·c
Lm
nm
fm
c
= Ltd +
#
!1
!1 + ! 2
$
·c
Ltd
fmod ↗ fmod→1
!1
fmod ↗ fmod+1
!2
c
=
=
>
>
>
>
G.O
n
n
↑
G.O = n x1 · x2 · · · · · xn
n
1 !
1!
ln(G.O) =
ln(xi ) =↓ G.O = e( n ln(xi ))
n
i=1
H.O
n
H.O = "n
1
i=1 xi
S2
S
n↗1
S2
S
n
1 !
S =
(xi ↗ X̄)2 ,
n↗1
2
S=
↑
S2
i=1
1
S2 =
n↗1
% n
!
i=1
x2i ↗ nX̄ 2
&
n
n
1 !
1 ! 2
(xi ↗ X̄)2 =
(xi ↗ 2xi X̄ + X̄ 2 )
n↗1
n↗1
i=1
i=1
% n
&
n
n
!
!
!
1
2
2
=
xi ↗ 2X̄
xi +
X̄
n↗1
S2 =
i=1
"
xi = nX̄
"
i=1
i=1
X̄ 2 = nX̄ 2
(
1 '! 2
xi ↗ 2X̄(nX̄) + nX̄ 2
n↗1%
&
% n
&
n
!
!
1
1
2
2
2
2
2
=
xi ↗ 2nX̄ + nX̄
=
xi ↗ nX̄
n↗1
n↗1
S2 =
i=1
i=1
m
1 !
1
S2 =
fi (yi ↗ Ȳ )2 =
n↗1
n↗1
i=1
%m
!
i=1
fi yi2 ↗ nȲ 2
&
Sp2
n
S12
m
Sp2 =
(n ↗ 1)S12 + (m ↗ 1)S22
n+m↗2
S22
↭
M.S
Sx̄
1 "n
M.S = n
i=1 |xi ↗ X̄|
S
Sx̄ = ± ↑
n
X̄ ± Sx̄
X
f (xi ) = P (X = xi )
f (xi ) → 0 (≃xi )
"N
i=1 f (xi ) = 1
E(X) = µX =
"N
i=1 xi f (xi )
F (x) = P (X ⇐ x) =
"
xk ↑x f (xk )
X
X ⇒ {1, 2, 3, 4, 5, 6}
E(X) =
6
!
i=1
f (xi ) = 16
# $
# $
# $
# $
# $
# $
1
1
1
1
1
1
21
xi f (xi ) = 1
+2
+3
+4
+5
+6
=
= 3.5
6
6
6
6
6
6
6
X
f (x)
f (x) → 0 (≃x ⇒ R)
) +↓
→↓ f (x) dx = 1
P (c < X < d) =
)d
c f (x) dx
P (X = c) = 0
)x
F (x) = P (X ⇐ x) = →↓ f (t) dt
) +↓
E(X) = µX = →↓ xf (x) dx
) +↓
V ar(X) = E(X 2 ) ↗ [E(X)]2 = →↓ x2 f (x) dx ↗ µ2X
a, b ⇒ R
X, Y
E(aX + b) = aE(X) + b
E(X + Y ) = E(X) + E(Y )
X
Y
E(XY ) = E(X)E(Y )
E(aX + b) = aE(X) + b
N
!
E(aX + b) =
(axi + b)f (xi ) =
i=1
N
!
=a
xi f (xi ) = E(X)
"
[axi f (xi ) + bf (xi )]
i=1
xi f (xi ) + b
i=1
"
N
!
N
!
f (xi )
i=1
f (xi ) = 1
E(aX + b) = aE(X) + b(1) = aE(X) + b ↭
E(X + Y ) = E(X) + E(Y )
* +↓ * +↓
E(X + Y ) =
(x + y)f (x, y) dx dy
→↓
=
=
→↓
* +↓ * +↓
→↓
* +↓
xf (x, y) dx dy +
→↓
xg(x) dx +
→↓
* +↓
→↓
* +↓ * +↓
→↓
yf (x, y) dx dy
→↓
yh(y) dy = E(X) + E(Y ) ↭
V ar(X) = E(X 2 ) ↗ [E(X)]2
V ar(aX) = a2 V ar(X)
V ar(X + b) = V ar(X)
V ar(X) = E(X 2 ) ↗ [E(X)]2
µ = E(X)
+
,
+
,
V ar(X) = E (X ↗ µ)2 = E X 2 ↗ 2µX + µ2
= E(X 2 ) ↗ E(2µX) + E(µ2 )
E(2µX) = 2µE(X) = 2µ2
E(µ2 ) = µ2
V ar(X) = E(X 2 ) ↗ 2µ2 + µ2 = E(X 2 ) ↗ µ2 = E(X 2 ) ↗ [E(X)]2
↭
V ar(aX) = a2 V ar(X)
+
,
+
,
V ar(aX) = E (aX ↗ E(aX))2 = E (aX ↗ aE(X))2
+
,
+
,
= E a2 (X ↗ E(X))2 = a2 E (X ↗ E(X))2 = a2 V ar(X) ↭
V ar(X + b) = V ar(X)
+
,
+
,
V ar(X + b) = E ((X + b) ↗ E(X + b))2 = E (X + b ↗ E(X) ↗ b)2
+
,
= E (X ↗ E(X))2 = V ar(X) ↭
→
→
a, b > 0
→
↑
↑
↑
↑
a+b ↑
( a ↗ b)2 → 0 =↓ a ↗ 2 ab + b → 0 =↓ a + b → 2 ab =↓
→ ab ↭
2
→
1
a
a
1
1
a + b
2
↑
→
→
→
ab →
-
b
1 1
a+b
1
· =↓
→↑
a b
2ab
ab
2ab
=↓
a+b
→
↭
1
b