February 2, 2007
HW1 Problem Solutions
#1.4.3.
First, remark that by defnition the random variables
V
j
=
U
j

U
j

1
are
iid
, where
j
≥
1 and
U
0
≡
0.
These variables each represent the
number o± trials up to an including the frst success, i.e. ±or
k, m
≥
1,
P
(
U
k

U
k

1
=
m
) =
P
(
V
1
=
m
) =
P
(
Y
1
=
Y
2
=
···
=
Y
m

1
, Y
m
= 1)
=
(1

p
)
m

1
p
All o± the
X
k
random variables are
iid
and independent o± the
iid
variables
U
j
as well.
We have to do three things: show that the variables
S
k
are mutually
independent, show that they all have the same distribution, and establish that
the exact ±orm o± the distribution o±
S
1
is exponenential as stated. For the frst
and second parts:
P
(
S
k
> t

U
k

1
=
n, X
1
, . . . , X
n
)
=
P
(
S
k
> t

U
k

1
=
n, X
1
, . . . , X
n
)
=
P
(
V
k
X
j
=1
X
n
+
j
> t

U
k

1
=
n
)
=
P
(
V
k
X
j
=1
X
j
> t
) =
P
(
S
1
> t
)
where the nexttolast step holds because
X
j
are
iid
and
X
’s are independent
o±
U
’s and
V
’s.
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 Spring '09
 SLUD

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