STAT 410
Examples for 07/17/2009
Summer 2009
Let
{
X
n
}
be a sequence of random variables and let
X
be a random variable.
Let
F
X
n
and
F
X
be, respectively, the c.d.f.s of
X n
and
X.
Let
C
(
F
X
)
denote the set of all
points where
F
X
is continuous.
We say that
X
n
converges in distribution
to
X
if
(
(
x
x
n
n
X
X
F
F
lim
=
∞
→
,
for all
x
∈
C
(
F
X
)
.
We denote this convergence by
X
X
D
n
→
.
Example 1
:
Consider
{
X
n
}
with
p.m.f.s
P
(
X
n
= 3
)
= 1 –
n
1
,
P
(
X
n
= 7
)
=
n
1
.
Note that
3
X
P
n
→
,
since
if
0 <
ε
< 4,
P
(

X
n
– 3

≥
ε
)
=
n
1
→
0
as
n
→
∞
,
and
if
ε
≥
4,
P
(

X
n
– 3

≥
ε
)
= 0.
Consider
X
with
p.m.f.
P
(
X = 3
)
= 1.
F
X
n
(
x
)
=
≥
<
≤

<
7
1
7
3
1
1
3
0
x
x
n
x
F
X
(
x
)
=
≥
<
7
1
3
0
x
x
(
(
x
x
n
n
X
X
F
F
lim
=
∞
→
,
for all
x
∈
R
.
⇒
X
X
D
n
→
.
Example 2
:
Let
{
X
n
}
, X
be
i.i.d.
with
p.m.f.
P
(
X
n
=
n
1
)
=
n
1
2
1

,
P
(
X
n
= 1
)
=
n
1
2
1
+
.
Then
F
X
n
(
x
)
=
≥
<
≤

<
1
1
1
1
1
2
1
1
0
x
x
n
n
n
x
.
( 29
≥
<
<
≤
∞
=
→
1
1
1
0
2
1
0
0
F
X
lim
x
x
x
x
n
n
.
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Consider
X
with
p.m.f.
P
(
X = 0
)
=
2
1
,
P
(
X = 1
)
=
2
1
.
Then
F
X
(
x
)
=
≥
<
≤
<
1
1
1
0
2
1
0
0
x
x
x
(
(
x
x
n
n
X
X
F
F
lim
=
∞
→
,
for all
x
≠
0.
F
X
n
(
0
)
= 0
for all
n
,
but
F
X
(
0
)
=
2
1
.
(
(
0
F
0
F
X
X
lim
≠
∞
→
n
n
.
Since
0
∉
C
(
F
X
)
,
X
X
D
n
→
.
Example 3
:
Suppose
P
(
X
n
=
i
)
=
6
3
+
n
i
n
,
for
i
= 1, 2, 3.
Then
X
X
D
n
→
,
where
P
(
X =
i
)
=
3
1
,
for
i
= 1, 2, 3.
Example 4
:
Let
X
n
have p.d.f.
f
n
(
x
)
=
n
x
n
– 1
,
for
0 <
x
< 1,
zero elsewhere.
Then
F
X
n
(
x
)
=
≥
<
≤
<
1
1
1
0
0
0
x
x
x
x
n
.
( 29
≥
<
∞
=
→
1
1
1
0
F
X
lim
x
x
x
n
n
.
Therefore,
X
X
D
n
→
,
where
P
(
X = 1
)
= 1.
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 Summer '08
 AlexeiStepanov

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