# Probabilidade e a Exercicios Com Gabarito

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DepartamentodeMatematicaSeccaodeEstatsticaeAplicacoesProbabilidadeseEstatsticaEXERCICIOSEdicao de Fevereiro de 2007FormularioP(X= x) =

nx

px(1 p)nxP(X= x) =exx!P(X= x) = p(1 p)x1x = 0, 1, . . . , n x = 0, 1, . . . x = 1, 2, . . .E(X) = np V ar(X) = np(1 p) E(X) = V ar(X) = E(X) =1pV ar(X) =(1 p)p2P(X= x) =

Mx

N Mn x

Nn

fX(x) =1b a,a x bx = max {0, n N+ M} , . . . , min {n, M} E(X) =b + a2V ar(X) =(b a)212E(X) = nMNV ar(X) = nMNN MNN nN 1fX(x) =122exp

(x )222,x IR fX(x) = ex,x 0E(X) = V ar(X) = 2E(X) =1V ar(X) =12X /n N(0, 1)X S/n t(n1)

X1X2

(12)

21n1+22n2 N(0, 1)X S/na N(0, 1) S2=1n 1ni=1

XiX

2

X1X2

(12)

S21n1+S22n2a N(0, 1)

X1X2

(12)

(n11)S21+(n21)S22n1+n22

1n1+1n2

t(n1+n22)(n 1)S22 2(n1)ki=1(OiEi)2Eia 2(k1)ri=1sj=1(OijEij)2Eija 2(r1)(s1)Yi = 0 + 1xi + i0=Y 1 x1=ni=1xiYin xYni=1x2i n x2 2=1n 2ni=1

YiYi

2,Yi =0 +1xi 2=1n 2ni=1Y2inY2

1

2

ni=1x2i n x200

1n+x2x2in x2

2 t(n2)11

2x2in x2 t(n2)

0 +1x0

(0 + 1x0)

1n+( xx0)2x2in x2

2 t(n2)R2=

ni=1xiYin xY

2

ni=1x2i n x2

ni=1Y2inY2