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THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT1301 Probability & Statistics I
Assignment 5
Due Date: December 5, 2011
(Hand in your solutions for Questions 3, 10, 13, 27, 29, 32, 34, 41)
1. Let
and
Y
be a continuous random variables distributed in
(
with pdf
(
1
,
0
~
U
X
)
)
1
,
0
( )
y
f
Y
and cdf
. Suppose
X
and
Y
are independent. Denote
as an indicator variable such
that
if A
occurs and
otherwise.
( )
y
F
Y
1
=
A
I
A
I
0
=
A
I
(a) Show that, for
,
1
0
<
<
t
t
y
t
y
I
I
y
t
y
Y
Y
t
X
P
<
≥
+
=
⎟
⎠
⎞
⎜
⎝
⎛
=
≤

.
(b) Using the result in part (a), show that the cdf of
XY
W
=
is given by
( )
( )
( )
⎪
⎪
⎩
⎪
⎪
⎨
⎧
≥
<
<
+
≤
=
∫
1
1
1
0
0
0
1
t
t
dy
y
f
y
t
t
F
t
t
F
t
Y
Y
W
.
(c) Using variable substitution and integral by parts for the integral in the expression of
in part (b), show that
( )
t
F
W
( )
t
F
W
can be also expressed as
( )
⎪
⎪
⎩
⎪
⎪
⎨
⎧
≥
<
<
⎟
⎠
⎞
⎜
⎝
⎛
+
≤
=
∫
1
1
1
0
0
0
1
t
t
dx
x
t
F
t
t
t
F
t
Y
W
.
(d) Derive the result in part (c) by first showing that
t
x
t
x
Y
I
I
x
t
F
x
X
X
t
Y
P
<
≥
+
⎟
⎠
⎞
⎜
⎝
⎛
=
⎟
⎠
⎞
⎜
⎝
⎛
=
≤

for
1
0
<
<
t
.
(e) Find the cdf of
if the cdf of
Y
is given by
XY
W
=
( )
⎪
⎩
⎪
⎨
⎧
≥
<
<
≤
=
1
1
1
0
0
0
2
y
y
y
y
y
F
Y
.
P. 1
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