Parte 5 (Aulas 9 e 10)  Valor Esperado e Variância
27 pág.

Parte 5 (Aulas 9 e 10) Valor Esperado e Variância


DisciplinaIntrodução à Estatística254 materiais2.035 seguidores
Pré-visualização1 página
Profa. Lidia Rodella 
UFPE-CAA 
 O valor esperado, , de uma variável 
aleatória discreta é definido como: 
)(XE
X
\uf0e5
\uf03d
\uf03d
n
i
ii xpxXE
1
)()(
onde, 
nxxx ,...,, 21
: valores possíveis de X. 
)()( ii xXPxp \uf03d\uf03d
\uf07d É a mesma média que aprendemos 
anteriormente? 
 
Notas de 100 alunos: 
 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,3,3,3,
3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,2,2,2,
2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,1,1,1,1,0 
Como podemos encontrar a média? 
- Somar todos os números e dividir por 100: 
92,2
100
292
\uf03d
- Multiplicar cada número pela quantidade de vezes que ele 
se repete, somar os resultados e dividir por 100: 
128324 \uf03d\uf0b4
105353 \uf03d\uf0b4
54272 \uf03d\uf0b4
551 \uf03d\uf0b4
010 \uf03d\uf0b4
2920554105128 \uf03d\uf02b\uf02b\uf02b\uf02b
92,2
100
292
\uf03d
x )(xp
4 
3 
2 
1 
0 
0,32 
0,35 
0,27 
0,05 
0,01 
)(xxp
1,28 
1,05 
0,54 
0,05 
0,00 
00,005,054,005,128,1)( \uf02b\uf02b\uf02b\uf02b\uf03dXE
92,2)( \uf03dXE
 é uma média ponderada dos diferentes 
valores de X com pesos dados pelas 
respectivas probabilidades. 
)(XE
\uf07d Ex: 
\u25e6 Experimento: lançamento de um dado; 
\u25e6 X: número de pontos obtidos. 
?)( \uf03dXE
x )(xp
1 
2 
3 
4 
5 
6 
1/6 
1/6 
1/6 
1/6 
1/6 
1/6 
6
1
6
6
1
5
6
1
4
6
1
3
6
1
2
6
1
1)( \uf0b4\uf02b\uf0b4\uf02b\uf0b4\uf02b\uf0b4\uf02b\uf0b4\uf02b\uf0b4\uf03dXE
5,3)( \uf03dXE
 não é o resultado que podemos esperar quando X 
for observada uma única vez. 
)(XE
 é a média aritmética dos resultados quando X for 
observada um grande número de vezes. 
)(XE
 O valor esperado, , de uma variável 
aleatória contínua é definido como: 
)(XE
X
\uf0f2
\uf02b\uf0a5
\uf0a5\uf02d
\uf03d dxxxfXE )()(
onde, 
)(xf
: é a função de densidade de probabilidade. 
\uf07d Ex: 
\uf03d)(xf
,2x
0, para outros valores de . 
10 \uf03c\uf03c x
x
\uf0f2
\uf02b\uf0a5
\uf0a5\uf02d
\uf03d dxxxfXE )()(
\uf0f2\uf0f2\uf0f2
\uf02b\uf0a5
\uf0a5\uf02d
\uf02b\uf02b\uf03d
1
1
0
0
)()()()( dxxxfdxxxfdxxxfXE
\uf0f2\uf0f2\uf0f2
\uf02b\uf0a5
\uf0a5\uf02d
\uf02b\uf02b\uf03d
1
1
0
0
020)( dxxdxxdxXE
\uf0f2 \uf03d\uf03d\uf03d
1
0
1
0
3
2
3
2
3
22)(
x
dxxXE
\uf07d Ex2: (relógio elétrico) 
\uf0f2 \uf03d\uf03d\uf03d
360
0
360
0
2
180
2360
1
360
1
)(
x
dxxXE
\uf03d)(xf
º0\uf03cx
º360º0 \uf03c\uf0a3 x
º360\uf0b3x
0, se 
1/360, se 
0, se 
a) Caso discreto: 
\uf0e5
\uf03d
\uf03d\uf03d
n
i
ii xpxHxHEYE
1
)()()]([][
X
b) Caso contínuo: 
Seja uma variável aleatória e seja 
).(XHY \uf03d
\uf0f2
\uf02b\uf0a5
\uf0a5\uf02d
\uf03d\uf03d dxxfxHxHEYE )()()]([][
X 0 1 2 
p(xi) 1/4 2/4 1/4 
a) Caso discreto: 
\uf0e5
\uf03d
\uf02b\uf03d\uf02b\uf03d
3
1
)()12(]12[)]([
i
ii xpxxExHE
b) Caso contínuo: 
\uf0f2\uf03d\uf03d
1
0
22 22]2[)]([ xdxxxExHE
Ex: 
12)( \uf02b\uf03d xxH
?)]([ \uf03dxHE
3
4
1
5
4
2
3
4
1
1 \uf03d\uf0b4\uf02b\uf0b4\uf02b\uf0b4\uf03d
\uf03d)(xf
,2x
0, para outros valores de . 
10 \uf03c\uf03c x
x
22)( xxH \uf03d
1|4 10
1
0
43 \uf03d\uf03d\uf03d \uf0f2 xdxx
Seja uma variável aleatória e uma constante. 
 
\uf07d ; 
 
\uf07d ; 
 
\uf07d ; 
 
 
 
 
Se então ; 
 
 
X C
CCE \uf03d)(
)()( XECCXE \uf0b4\uf03d
bXaY \uf02b\uf03d )()( XbEaYE \uf02b\uf03d
CXECXE \uf0b1\uf03d\uf0b1 )()(
Seja uma variável aleatória e uma constante. 
 
\uf07d k funções de , 
então 
X C
:)(),...,(),( 21 XHXHXH K X
)]([...)]([)]([)](...)()([ 2121 XHEXHEXHEXHXHXHE KK \uf02b\uf02b\uf02b\uf03d\uf02b\uf02b\uf02b
\uf07d Ex: 
x p(x) 
1 ¼ 
2 ½ 
3 ¼ 
4
1
3
2
1
2
4
1
1)( \uf0b4\uf02b\uf0b4\uf02b\uf0b4\uf03dXE
2\uf03d
x+3 p(x) 
¼ 
½ 
¼ 
4 
5 
6 
CXECXE \uf02b\uf03d\uf02b )()(
532 \uf03d\uf02b\uf03d
4
1
6
2
1
5
4
1
4)3( \uf0b4\uf02b\uf0b4\uf02b\uf0b4\uf03d\uf02bXE
5\uf03d
\uf07d Ex: 
x p(x) 
1 ¼ 
2 ½ 
3 ¼ 
4
1
3
2
1
2
4
1
1)( \uf0b4\uf02b\uf0b4\uf02b\uf0b4\uf03dXE
2\uf03d
2x p(x) 
¼ 
½ 
¼ 
2 
4 
6 
)()( XECCXE \uf0b4\uf03d
422 \uf03d\uf0b4\uf03d
4
1
6
2
1
4
4
1
2)2( \uf0b4\uf02b\uf0b4\uf02b\uf0b4\uf03dXE
4\uf03d
\uf07d Ex: 
x p(x) 
1 ¼ 
2 ½ 
3 ¼ 
4
1
3
2
1
2
4
1
1)( \uf0b4\uf02b\uf0b4\uf02b\uf0b4\uf03dXE
2\uf03d
2x+1 p(x) 
¼ 
½ 
¼ 
3 
5 
7 
1)(2)12( \uf02b\uf03d\uf02b XEXE
5122 \uf03d\uf02b\uf0b4\uf03d
4
1
7
2
1
5
4
1
3)12( \uf0b4\uf02b\uf0b4\uf02b\uf0b4\uf03d\uf02bXE
5\uf03d
\uf07d Ex: 
\uf03d)2( 2XE 1
2
1
2 \uf03d\uf0b4\uf03d
\uf03d)(xf
,2x
0, para outros valores de . 
10 \uf03c\uf03c x
x
22)( xxH \uf03d
?)]([ \uf03dxHE
\uf03d)( 2XE
)(2 2XE
\uf0f2
1
0
22xdxx
1
0
4
2
x
\uf03d
2
1
\uf03d
a) Caso discreto: 
\uf0e5
\uf03d
\uf02d\uf03d
n
i
ii xpXExXV
1
2 )()]([][
b) Caso contínuo: 
2)]([)( XEXEXV \uf02d\uf03d
\uf0f2
\uf02b\uf0a5
\uf0a5\uf02d
\uf02d\uf03d dxxfxExXV )()]([][ 2
\uf07d Ex: 
x p(x) 
1 ¼ 
2 ½ 
3 ¼ 
2)( \uf03dXE
\uf0e5
\uf03d
\uf02d\uf03d
3
1
2 )()2()(
i
ii xpxXV
\uf0e5
\uf03d
\uf02d\uf03d
n
i
ii xpXExXV
1
2 )()]([][
2
1
4
1
10
4
1
1 \uf03d\uf0b4\uf02b\uf02b\uf0b4\uf03d
a) Caso discreto: 
\uf07d Ex: 
\uf03d)(xf
,2x
0, para outros valores de . 
10 \uf03c\uf03c x
x
b) Caso contínuo: 
3
2
)( \uf03dXE
\uf0f2
\uf02b\uf0a5
\uf0a5\uf02d
\uf02d\uf03d dxxfxExXV )()]([][ 2
\uf0f2
\uf02b\uf0a5
\uf0a5\uf02d
\uf02d\uf03d dxxfxXV )(]
3
2
[][ 2 \uf03d\uf02b\uf02d\uf03d \uf0f2
\uf02b\uf0a5
\uf0a5\uf02d
dxxfxx )()
9
4
3
4
( 2
\uf03d\uf02b\uf02d\uf03d \uf0f2 \uf0f2\uf0f2
\uf02b\uf0a5
\uf0a5\uf02d
\uf02b\uf0a5
\uf0a5\uf02d
\uf02b\uf0a5
\uf0a5\uf02d
dxxfdxxxfdxxfx )(
9
4
)(
3
4
)(2
\uf03d\uf0b4\uf02b\uf02d\uf03d \uf0f2
1
0
2 1
9
4
)(
3
4
2 XExdxx \uf03d\uf02b\uf0b4\uf02d
9
4
3
2
3
4
2
1
0
4x
18
1
9
4
2
1
\uf03d\uf02d
\uf07d Teorema: 
)()()( 22 XEXEXV \uf02d\uf03d
2)]([)( XEXEXV \uf02d\uf03d
Demonstração: 
)]()(2[ 22 XEXXEXE \uf02b\uf02d\uf03d
)]([)](2[)( 22 XEEXXEEXE \uf02b\uf02d\uf03d
)()()(2)( 22 XEXEXEXE \uf02b\uf02d\uf03d
)()(2)( 222 XEXEXE \uf02b\uf02d\uf03d
)()( 22 XEXE \uf02d\uf03d
\uf07d Ex: 
x p(x) 
1 ¼ 
2 ½ 
3 ¼ 
2)( \uf03dXE
\uf0e5
\uf03d
\uf02d\uf03d
3
1
2 )()2()(
i
ii xpxXV 2
1
4
1
10
4
1
1 \uf03d\uf0b4\uf02b\uf02b\uf0b4\uf03d
a) Caso discreto: 
)()()( 22 XEXEXV \uf02d\uf03d
\uf0e5
\uf03d
\uf03d
3
1
22 )()(
i
i xipxXE
2
9
2
5
2
4
1
9
2
1
4
4
1
1 \uf03d\uf02b\uf03d\uf0b4\uf02b\uf0b4\uf02b\uf0b4\uf03d
2
1
2
2
9 2 \uf03d\uf02d\uf03d
\uf07d Ex: 
\uf03d)(xf
,2x
0, para outros valores de . 
10 \uf03c\uf03c x
x
b) Caso contínuo: 
3
2
)( \uf03dXE
18
1
)()]([][ 2 \uf03d\uf02d\uf03d \uf0f2
\uf02b\uf0a5
\uf0a5\uf02d
dxxfxExXV
)()()( 22 XEXEXV \uf02d\uf03d
\uf0f2\uf03d
1
0
22 2)( xdxxXE
2
1
2
1
0
4
\uf03d\uf03d
x
18
1
3
2
2
1
2
\uf03d\uf0f7
\uf0f8
\uf0f6
\uf0e7
\uf0e8
\uf0e6
\uf02d\uf03d
Seja uma variável aleatória e uma constante. 
 
\uf07d ; 
 
\uf07d ; 
 
 
 
 
 
X C
0)( \uf03dCV
)()( 2 XVCCXV \uf0b4\uf03d
Seja uma variável aleatória e uma constante. 
 
 
 
X C
).()( XVCXV \uf03d\uf0b1
\uf07d Ex: 
x p(x) 
1 ¼ 
2 ½ 
3 ¼ 
2
1
)( \uf03dXV
x+3 p(x) 
¼ 
½ 
¼ 
4 
5 
6 
\uf03d\uf02b )3(XV
2
1
4
1
)56(
2
1
)55(
4
1
)54()3( 222 \uf03d\uf0b4\uf02d\uf02b\uf0b4\uf02d\uf02b\uf0b4\uf02d\uf03d\uf02bXV
2
1
)( \uf03dXV
5)3( \uf03d\uf02bXE
2)( \uf03dXE
\uf07d Ex: 
x p(x) 
1 ¼ 
2 ½ 
3 ¼ 
2)( \uf03dXE
2x p(x) 
¼ 
½ 
¼ 
2 
4 
6 
)2( XV
2
2
1
4 \uf03d\uf0b4\uf03d
4)2( \uf03dXE
2
1
)( \uf03dXV
)(22 XV\uf0b4\uf03d
2
4
1
)46(
2
1
)44(
4
1
)42()2( 222 \uf03d\uf0b4\uf02d\uf02b\uf0b4\uf02d\uf02b\uf0b4\uf02d\uf03dXV
1. Uma moeda perfeita é lançada 3 vezes. 
Seja Y o número de caras obtidas. 
 
 
 
 
 
 
 
Calcule o valor esperado e a variância. 
 
Y p(y) 
0 1/8 
1 3/8 
2 3/8 
3 1/8