2009 Fall Stat 426 : Homework 1
Moulinath Banerjee
University of Michigan
Announcement:
The homework carries 50 points and contributes 5
points to the total grade. Your score on the homework is scaled down to
out of 5 and recorded.
1. A geometric random variable
W
takes values
{
1, 2, 3, .
. .
}
and
P
(
W
=
j
) =
θ
(1

θ
)
j

1
,where 0
< θ <
1.
(a) Prove that for any two positive integers
i
,
j
, it is the case that,
P
(
W > i
+
j

W > i
) =
P
(
W > j
) .
(b) Indeed, the converse is also true. We show that if
W
is a discrete
random variable taking values
{
1
,
2
,
3
,
. . .
}
with probabilities
{
p
1
,p
2
,p
3
,
. . .
}
and satisﬁes the memoryless property, then
W
must follow a geometric distribution.
Follow these steps to establish the fact that
W
is geometric. Using
the fact that
W
has the memoryless property, show that
P
(
W > m
) = (
P
(
W >
1))
m
,
for any
m
≥
2. As a ﬁrst step towards proving this show that
P
(
W >
2) = (
P
(
W >
1))
2
Deﬁne
θ
=
P
(
W
= 1) and 1

θ
=
P
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 Winter '08
 moulib
 Statistics, Probability theory, memoryless, Moulinath Banerjee University of Michigan

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