→
→
→
→
→
→
n
∑
Xi
̅
X = i=1
n
n
̅ = ∑i=1 Fi Xi
X
n
̅
X = ∑ni=1 fi Xi
fi =
Fi
n
→
→
→
S2 =
S2 =
2
S =
2
̅ 2
∑n
i=1(Xi −X)
S ′2 =
n
2
̅2
∑n
i=1 Xi −nX
n
̅ 2
∑n
i=1 Fi (Xi −X)
n
∑n
2
̅2
̅ 2
∑n
i=1(Xi −X)
n−1
s = √S 2 = √
CV =
̅) 2
∑ni=1(Xi − X
n
s
∗ 100
̅
X
g1 =
n2 M3
(n−1)(n−2)S
M3 =
′3
̅ 3
∑n
i=1(Xi −X)
n
→
→
→
g2 =
n2 (𝑛+1)M4
(n−1)(n−2)(𝑛−3)S′
4
4
M =
̅ 4
∑n
i=1(Xi −X)
n
S
′4
̅ 2
∑n
i=1(Xi −X)
= [√
n−1
4
]
3
S ′ = [√
̅ 2
∑n
i=1(Xi −X)
n−1
3
]
Número de semanários lidos por mês Gráfico de Ramos e Folhas
Frequência
Raiz &
2,00
2
5,00
3
13,00
4
16,00
5
17,00
6
11,00
7
10,00
8
7,00
9
8,00
10
,00
11
4,00
12
,00
13
3,00
14
2,00
15
2,00 Extremos
Largura do ramo:
Cada folha:
Folha
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(>=16)
1
1 caso(s)
Ω
Ω
Ω
⊂
P[X = x] se x = xj
f(x) = {
0 se x ≠ xj
∑ f(xi ) = 1
F(x) = P[X ≤ x]
∀ ∈
∀
∈
lim F(x) = 0
x→−∞
lim F(x) = 1
x→+∞
P[x1 < X ≤ x2 ] = F(x2 ) − F(x1 ) ∀
∈
X
0
1
f(x) 0,25 0,75
F(x) 0,25
1
P[X = x] = 0
f(x) ≥ 0
+∞
∫−∞ f(x)dx = 1
x
F(x) = P[X ≤ x] = ∫ f(x)dx
−∞
b
P[a ≤ X ≤ b] = ∫ f(x)dx = F(b) − F(a)
a
P[A/B] =
P[A ∩ B]
P[B]
E[X] = μ
∗ Var[X] = σ = E[(X − μ)2 ] = E[X 2 ] − (E[X])2 ∗
2
+∞
E[X] = ∫
𝐄[𝐗] = ∑ 𝐱 𝐢 𝐟(𝐱 𝐢 )
x. f(x)dx
−∞
+∞
𝐢
𝐕𝐚𝐫[𝐗] = ∑ (𝐱 𝐢 − 𝛍)𝟐 𝐟(𝐱𝐢 )
(x − μ)2 f(x)dx
Var[X] = ∫
𝐢
−∞
Var[K] = 0
Var[KX] = K 2 Var[X]
E[K] = K
E[KX] = KE[X]
E[X ± Y] = E[X] ± E[Y]
E[XY] = E[X]. E[Y]
∗ Var[X] = σ2 = E[(X − μ)2 ] = E[(X 2 − 2μX + μ2 )] =
E[X 2 ] − 2μE[X] + E[X 2 ] − μ2 = E[X 2 ] − (E[X])2 ∗
Var[aX ± bY] = 𝑎2 Var[𝑋] + 𝑏 2 Var[𝑌] ± 2 ∗
Cov[X, Y]
+∞
𝐄[𝐠(𝒙𝒊 )] = ∑ 𝐠(𝐱 𝐢 )𝐟(𝐱 𝐢 )
E[𝑔(𝑥)] = ∫
𝐢
−∞
Ω
f(x, y) = P[X = x, Y = y]
0 ≤ f(x, y) ≤ 1 ∀ (
∑x ∑y f(x, y) = 1
∈
fX (x) = ∑y f(x, y) = P[X = x, −∞ < Y < +∞]
f𝑌 (𝑦) = ∑x f(x, y) = P[−∞ < X < +∞, Y = y]
f(x, y) = fX (x). f𝑌 (𝑦) ∀ (
g(x). f(x)dx
r
s
F(x, y) = P[X ≤ x, Y ≤ y] = ∑ ∑ f(xi , yj ) = ∑ ∑ f(xi , yj )
i=1 j=1
∀
∀
lim
x/y→−
∞
F(x, y) = 0
lim
F(x, y) = 0
lim
F(x, y) = 1
x→−∞ e y→−∞
x→+∞ e y→+∞
∀ ∈
→
∀
Cov[X, Y] = E[(X − μx )(Y − μy )]
= E[(XY − μx Y − μy X + μx μy )]
= E[XY] − μx E(Y) − μy E(X) + μx μy
= E[XY] − E[X]E[Y]
ρxy =
ρxy
ρxy = −1 →
ρxy = 1 →
ρxy = 0 →
Cov[X, Y]
√Var[X] ∗ Var[Y]
xi ≤x yj ≤y
1
x = 1, 2, … , n
f(x) = {n ,
0
outros valores
n+1
n2 − 1
E[X] =
e Var[X] =
2
12
→
→
x
f(x) = P[X = x] = {p ∗
0
(1 − p)1−x ,
x = 0, 1
outros valores
Var[X] = p ∗ q = p ∗ (1 − p)
E[X] = p e
→
∩
n
( ) ∗ px (1 − p)n−x , x = 0, 1, … , n
f(x) = P[X = x] = { x
0
outros valores
= 𝑝 ∗ 𝑝 ∗ 𝑝 ∗ …∗ 𝑞 ∗ 𝑞 ∗ 𝑞
𝑓(𝑥) = 𝑃[𝑋 = 𝑥] =
= 𝑝(1 − p)n−1
∩
E[X] = n ∗ p e
Var[X] = n ∗ p ∗ q = n ∗ p ∗ (1 − p)
λ
e−λ ∗ λx
, x = 0, 1, …
f(x) = P[X = x] = { x!
0
outros valores
𝑛!
𝑛
( )=
𝑥
(𝑛 − 𝑥)! 𝑥!
E[X] = λ e
Var[X] = λ
∩
∑𝐾
𝑖=1 𝑋𝑖 ∩
λ𝑖
→
→
∑𝐾
𝑖=1 λ𝑖
λ = n∗p
X ∩ P(λ = n. p)
∩
1
𝑓(x) = { b − a a < x < b
0 outros valores
−∞ < 𝑎 < 𝑏 < +∞
0
x≤a
x−a
a<x<b
F(x) = {
b−a
1
x≥b
a+b
(𝑏 − 𝑎)2
E[X] =
e Var[X] =
2
12
∩
f(x) =
1
e
1 x−μ 2
− (
)
2 σ
σ√2π
E[X] = μ e
𝜇, 𝜎
, −∞<x<+∞
Var[X] = σ2
𝜇
𝜇±𝜎
𝜇
−∞<μ<+∞
𝜎
𝜎>0
𝜇
𝜎
𝜇 = 0, 𝜎 = 1 →
∩
𝐸൤
X−μ
1
൨ = 𝐸[𝑋 − 𝜇] =
σ
𝜎
1
= (𝐸[𝑋] − 𝜇) = 0
𝜎
𝐸[𝑋] = 𝜇
𝜇
X−μ
1
𝑉𝑎𝑟 ൤
൨ = 2 𝑉𝑎𝑟[𝑋 − 𝜇] =
σ
𝜎
1
= 2 (𝑉𝑎𝑟[𝑋] − 0) = 1
𝜎
0
𝐙=
𝐗−𝛍
𝛔
∩
𝜇 𝜎
T = ∑ni ai Xi ∩ N (μ = ∑ni ai μi ; σ = √∑ni a2i σ2i )
∩
∑ni X i
∩
𝜇 𝜎
∩
𝜇 𝜎
∩ N(n ∗ μ; σ√n)
𝜇 𝜎
X1 + X2 ∩ N (μ1 +μ2 ; √σ12 + σ22 )
X1 − X2 ∩ N (μ1 −μ2 ; √σ12 + σ22 )
∩
𝜇 𝜎
σ
)
√n
∑𝑛
𝑖 𝑋𝑖
̅ ∩ N (μ;
X
̅
X=
𝑛
→
→
X ∩ N(μ = n. p; σ = √𝑛𝑝𝑞 = √𝑛𝑝(1 − 𝑝))
𝝀→
𝜆
𝜆
X ∩̇ N(μ = 𝜆; σ = √𝜆)
≠ →
→
→
X ∩ χ2(n)
E [χ2(n) ] = n
Var [χ2(n) ] = 2n
X ∩ N(μ, σ)
X−μ 2
) ∩ χ2(1) →
Z =(
σ
2
Xi
Xi ∩ N(μi, σi)
Xi − μi 2
) ∩ χ2(n) →
∑ (
σi
i=1
n
X ∩ χ2(m)
Y ∩ χ2(n)
X + Y ∩ χ2(m+n) →
2
𝜒(𝑛)
→
χ2(n) ∩ N(n, √2n)
X ∩ t (n)
X ∩ N(μ, σ)
Y ∩ χ2(n)
X−μ
σ ∩t
(n)
Y
√
n
X ∩ t (n)
E[X] = 0
n
Var[X] =
, com n > 2
n−2
→
t (n)
𝑛
)
𝑛−2
t (n) ∩ N (0, √
X ∩ F(m,n)
𝑛
𝑛−2
2𝑛2 (𝑚 + 𝑛 − 2)
Var [χ2(n) ] =
𝑚(𝑛 − 2)2 (𝑛 − 4)
E[X] =
χ2(m)
χ2(n)
F(m,n)
X ∩ χ2(m)
Y ∩ χ2(n)
X
m ∩F
(m,n)
Y
n
𝑃[𝐹(10,5) < 𝑎] = 0,05
1 − 𝑃[𝐹(10,5) ≥ 𝑎] = 0,05
𝑃[
X ∩ F(m,n)
1
∩ F(n,m)
X
1
1
1
> ] = 0,05 ⇔ 𝑃 ൤𝐹(5,10) > ൨ = 0,05
𝐹(10,5) 𝑎
𝑎
1
⇔ 𝑃 ൤𝐹(5,10) ≤ ൨ = 0,95
𝑎
T ∩ t (n)
T 2 ∩ F(1,n)
X ∩ N(μ, σ)
X−μ
σ
Y
n
√
∩ t (n)
χ2(1)
X−μ 2
X−μ 2
( σ )
( σ )
2 =
Y
Y
n
(√n)
χ2(n)
≠
→
→
→
≠
(X1 , X 2 , … , Xn )
f(x1 , x2 , … , xn )
(X1 , X2 , … , Xn )
(X1 , X2 , … , Xn )
f(x1 , x2 , … , xn ) = f(x1 ) ∗ f(x2 ) ∗ … ∗ f(xn )
Y ∩ χ2(n)
→
(X1 , X2 , … , Xn )
E[Xi] = μ
n
Xi ∩̇ N(nμ; σ√n) →
∑
∑ni=1 Xi − nμ
σ√n
i=1
∩̇ N(0, 1)
∑ni=1 Xi
∑ni=1 Xi
∑ni=1 Xi
−
μ
−
μ
̅−μ
σ
n
n −μ = X
̅ ∩̇ N (μ, )
= n
=
∩̇ N(0, 1) → X
σ
σ
n
σ√n
√n
σ√ 2
√n
√n
n
n
f(x) = px ∗ (1 − p)1−x ,
x = 0, 1
∑ni=1 Xi
n
∑
Xi ∩ b(n, p)
i=1
n
Xi ∩̇ N(np; √npq) →
∑
∑ni=1 Xi − np
i=1
(∑n
i=1 Xi)
n
√npq
∩̇ N(0, 1)
̅
=X
∑ni=1 Xi
̅
np
pq
n − p = X − p ∩̇ N(0,1) → X
̅ ∩̇ N (p, √ )
=
n
pq
√npq
√npq
√
n
n
∑ni=1 Xi −
(X1 , X2 , … , Xn )
𝜇
𝜎
n
̅ = (∑i=1 Xi)
X
n
Xi ∩ N(μ, σ)
̅
X ∩ N(μ, σ/√𝑛)
Var[Xi] = σ2
S2 =
̅ )2
∑(Xi−X
n
(X1 , X2 , … , Xn )
Xi − μ
∩ N(0,1)
σ
2
n
Xi − μ 2
Xi − ̅
X
2
∑ (
) ∩ χ(n) → ∑ (
) ∩ χ2(n−1)
σ
σ
i=1
i=1
n
̅)2
̅)2 nS2
n(Xi − X
n ∑(Xi − X
∑
= 2
= 2 ∩ χ2(n−1)
2
nσ
σ
n
σ
i=1
n
S′2 =
̅ )2
∑(Xi−X
n−1
nS2 = (n − 1)S′
2
2
(n−1)S′
σ2
∩ χ2(n−1)
θ̂
θ̂
θ̂
θ
E[θ̂] = θ
→
E[θ̂] − θ
→
θ̂
θ
θ̂
θ̃
Var(θ̂) ≤ Var(θ̃)
θ̂
θ̂
θ̃
θ
→
θ̂n
E[𝑆 2 ] =
n−1
n
∗ 𝜎2 →
lim E[θ̂n ] = θ
𝑛→∞
𝜎2
lim
n−1
𝑛→∞ n
∗ 𝜎2 = 𝜎2 →
𝜎2
θ̂n
lim EQM( θ̂n ) = lim [Var(θ̂n ) + (env. (θ̂n ))2 ] = 0
n→∞
n→∞
lim [E[θ̂n ]] = θ
n→∞
lim [Var(θ̂n )] = 0
n→∞
θ̂n