STAT 410
Homework #10
Spring 2009
(due Friday, July 31, by 4:00 p.m.)
1.
Suppose
X
1
, X
2
, … , X
n
are independent random variables,
and
X
i
has
Geometric distribution with probability of “success”
p
i
,
i
= 1, 2, … ,
n
.
Let
Y =
min X
i
.
What is the probability distribution of
Y?
Hint:
Consider
P
(
X >
x
)
for a
Geometric
(
p
)
random variable.
Let
X
be a random variable with a Geometric distribution with probability of
“success”
p
.
Then
P
(
X >
y
)
=
( 29
∑
∞

+
=

⋅
1
1
1
y
k
k
p
p
=
(
1 –
p
)
y
,
y
= 0, 1, 2, 3, … .
Let
y
be a positive integer.
P
(
Y >
y
)
=
P
(
X
1
>
y
)
⋅
P
(
X
2
>
y
)
⋅
…
⋅
P
(
X
n
>
y
)
=
(
1 –
p
1
)
y
⋅
(
1 –
p
2
)
y
⋅
…
⋅
(
1 –
p
n
)
y
=
( 29
y
n
i
i
p
1
1

∏
=
,
y
= 0, 1, 2, 3, … .
p
Y
(
y
)
=
P
(
Y =
y
)
=
P
(
Y >
y
– 1
)
–
P
(
Y >
y
)
=
( 29 ( 29
1
1
1
1
1
1

=
=



∏
∏
⋅
y
n
i
i
n
i
i
p
p
,
y
= 1, 2, 3, … .
Y
has a
Geometric distribution with probability of “success”
p
=
( 29
∏
=


n
i
i
p
1
1
1
.