iausha

´
Departamento de Matematica

Probabilidades e Estat´ ıstica Exerc´ ıcios Revisto em Fevereiro de 2010 por Carlos Daniel Paulino, Paulo Soares e Giovani Silva.

Formul´rio a P (X = x) =

n x p (1 − p)n−x x P (X = x) =

x = 0, 1, . . . , n

x = 1, 2, . . .

E(X) = V ar(X) = λ

N −M n−x M x P (X = x) =

N n M
N

V ar(X) = n

fX (x) = √

S2 =

2

¯
Xi − X

E(X) =

(n − 1)S 2
∼ χ2
(n−1)
σ2

1 λ V ar(X) =

¯
¯
X1 − X2 − (µ1 − µ2 )
2
σ1 n1 +

¯
¯
X1 − X2 − (µ1 − µ2 )

a

∼ N (0, 1)

2
S2
n2

2
2
(n1 −1)S1 +(n2 −1)S2 n1 +n2 −2

(Oi − Ei )2 a 2
∼ χ(k−m−1)
H0
Ei i=1 1 n1 r

s

ˆ
ˆ¯
¯ β0 = Y − β1 x

Y i = β 0 + β 1 xi + εi

ˆ β1 =

xi Yi − n¯Y x¯ i=1 n i=1 n 1
ˆ
ˆ
ˆ 2 ˆ σ =
ˆ
Yi − Yi , Yi = β0 + β1 xi n − 2 i=1

+

x2 x2 −n¯2 x i

σ2
ˆ

ˆ β1 − β1

i=1 i=1 x2 i i=1

ˆ
¯
− nY 2 − β1

1 n xi Yi − n¯Y x¯ − n¯2 × x x2 − n¯2 x i
2

n i=1 x2 − n¯2 x i

ˆ
ˆ
β0 + β1 x0 − (β0 + β1 x0 )

∼ t(n−2)

σ2
ˆ
x2 −n¯2 x i

n

Yi2

2

n

R2 =

n

1 σ =
ˆ
n−2

2

∼ t(n−2)

∼ t(n1 +n2 −2)

1 n2 +

n

1 n ∼ N (0, 1)

(Oij − Eij )2 a 2
∼ χ(r−1)(s−1)
H0
Eij i=1 j=1

k

ˆ β0 − β0

2 σ2 n2

1 λ2 n−1

¯
¯
X1 − X2 − (µ1 − µ2 )
+

fX (x) = λe−λx , x ≥ 0

V ar(X) = σ 2

n i=1 (1 − p) p2 V ar(X) =

M N −M N −n
N
N N −1

¯
X −µ
√ ∼ t(n−1)
S/ n

¯
X −µ
√ ∼ N (0, 1) σ/ n

1 p 1
,a≤x≤b
b−a b+a (b − a)2
E(X) =
V ar(X) =
2
12

1
(x − µ)2 exp −
,x∈I
R
2σ 2
2πσ 2

E(X) = µ

2
S1
n1

E(X) =

fX (x) =

x = max {0, n − N + M } , . . . , min {n, M }
E(X) = n

P (X = x) = p(1 − p)x−1

x = 0, 1, . . .

V ar(X) = np(1 − p)

E(X) = np

e−λ λx x! n i=1 2

¯
Yi2 − nY 2

+

(¯−x0 )2 x x2 −n¯2 x i

σ2
ˆ

∼ t(n−2)

Cap´ ıtulo 1
Estat´
ıstica descritiva
1.1 Uma