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Statistics 330/600
Due: Thursday 7 April
2005: Sheet 10
*(10.1)
Suppose
X
∈
L
1
(Ä,
F
,
P
)
and
Y
=
P
(
X

F
0
)
for some subsigmaFeld
F
0
of
F
. Suppose
W
is
an
F
0
measurable random variable for which
XW
∈
L
1
(Ä,
F
,
P
)
.
(i) Show that
YW
∈
L
1
(Ä,
F
,
P
)
and
P
(
XW
)
=
P
(
YW
)
. Hint: Consider
X
±
and
W
±
. Note
that
Y
=
P
(
X
+

F
0
)
−
P
(
X
−

F
0
)
almost surely.
(ii) Show that
Y
≥
0 almost surely if
P
(
XW
)
≥
0 for every bounded,
F
0
measurable,
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This note was uploaded on 11/21/2009 for the course STAT 330 taught by Professor Davidpollard during the Spring '09 term at Yale.
 Spring '09
 DavidPollard
 Statistics, Probability

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