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3UDZGRSRGRELH
VWZR L VWDW\VW\ND
16.11.1996 r.
___________________________________________________________________________
Zadanie 1.
:LDGRPR
*H$
%L&V
WU]HPD ]GDU]HQLDPL ORVRZ\PL WDNLPL
*H
()
5
A
=
2
,
Pr
()
4
A
=
1
,
Pr
C
A
Ç
B
)
=
1
,
2
Pr
(
A
È
B
)
=
6
,
Pr
()
3
B
=
1
.
10
(
)
Ile wynosi
Pr
A
B
Ç
C
?
(A)
2
3
(B)
3
5
(C)
1
2
(D)
3
10
(E)
3RGDQH LQIRUPDFMH QLH Z\VWDUF]DM
GR XG]LHOHQLD RGSRZLHG]L
1
Pr
B
(
C
3UDZGRSRGRELH
VWZR L VWDW\VW\ND
16.11.1996 r.
___________________________________________________________________________
Zadanie 2.
Niech
U
1
,
2
U
,
,
U
n
E
G]LH SUyE
] UR]NÆDGX MHGQRVWDMQHJR QD SU]HG]LDOH
()
{
}
a
,
b
5R]ZD*P\ ]PLHQQH ORVRZH
X
=
min
U
1
,
U
2
,
,
U
n
oraz
{
}
( )
Y
=
max
U
1
,
U
2
,
,
U
n
:VSyÆF]\QQLN NRUHODFML OLQLRZHM
Corr
, w i:
X
Y
(A)
-
1
n
(B)
1
n
(C)
n
1
-
1
(
1
b
n
a
2
(D)
-
(E)
0
2
3UDZGRSRGRELH
VWZR L VWDW\VW\ND
16.11.1996 r.
___________________________________________________________________________
X
,
E
G]LH SUyE
QLH]DOH*Q\FK UHDOL]DFML ]
GZXZ\PLDURZHJR UR]NÆDGX QRUPDOQHJR R ZHNWRU]H ZDUWR
FL RF]HNLZDQ\FK
(
X
1
,
Y
1
)
, ... ,
(
n
Y
n
)
(
)
m
X
,
m
Y
i macierzy kowariancji
é
1
r
r
.
ù
ë
û
1
ì
1
×
åå
=
n
1
n
ü
Statystyka
min
X
,
×
Y
n
1
i
n
i
i
i
=
1
{
}
MHVW QLHREFL
*RQ\P HVW\PDWRUHP SDUDPHWUX
min
m
X
,
m
Y
:
(A)
GOD ND*GHJR
r
(B)
W\ONR
MH
OL
r
=
0
(C)
W\ONR
MH
OL
r
=
1
(D)
W\ONR
MH
OL
r
=
1
(E)
nigdy
3
Zadanie 3.
Niech
3UDZGRSRGRELH
VWZR L VWDW\VW\ND
16.11.1996 r.
___________________________________________________________________________
Zadanie 4.
Zmienna losowa
X
PD UR]NÆDG ORJDU\WPLF]QR
QRUPDOQ\ R J
VWR
FL
SUDZGRSRGRELH
VWZD
Ã
1
×
exp
é
-
1
×
(
ln
x
-
m
)
2
ù
dla
x
>
0
f
X
(
x
)
=
2
p
x
ë
2
û
î
0
poza
tym
(
)
(
)
:LDGRPR
*H
Pr
X
£
q
=
0
6
oraz
Pr
X
£
r
=
0
:\QLND VW
G
*H
(A)
E
()
X
=
q
×
r
×
e
(B)
E
( )
ln
X
=
q
×
r
(C)
3RGDQH LQIRUPDFMH V
VSU]HF]QH
(D)
E
()
X
=
q
+
r
2
(E)
E
()
X
=
q
×
r
4
ì
3UDZGRSRGRELH
VWZR L VWDW\VW\ND
16.11.1996 r.
___________________________________________________________________________
Zadanie 5.
Zmienne losowe
X
1
,
2
X
,
,
X
n
V
QLH]DOH*QH L PDM
MHGQDNRZ\ UR]NÆDG
Z\NÆDGQLF]\ R J
VWR
FL
ì
e
-
x
dla
x
>
0
f
(
x
)
=
0
poza
tym
î
Zmienna losowa
N
MHVW QLH]DOH*QD RG QLFK L PD UR]NÆDG JHRPHWU\F]Q\
(
N
=
k
) ( )
=
1
-
q
×
q
k
k
=
0
1
Ã¥
=
N
Niech
S
=
X
i
E
G]LH VXP
ORVRZHM OLF]E\ ]PLHQQ\FK ORVRZ\FK
SU]\MPXMHP\ *H
i
1
Ã¥
=
0
X
i
=
0
).
i
1
’OD ND*GHJR
0
s
SUDZG
MHVW
*H
>
(A)
Var
(
N
S
=
s
) (
=
E
N
S
=
s
)
-
1
(B)
Var
(
N
S
=
s
) (
=
E
N
S
=
s
)
(C)
Var
(
N
S
=
s
)
=
Var
()
N
-
[
E
(
N
S
=
s
]
2
(D)
Var
(
N
S
=
s
)
=
s
×
q
2
(E)
Var
(
N
S
=
s
)
=
s
×
q
5
Pr
)
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