1 REVIEW OF INDEPENDENCE RANDOM VECTORS
Lecture 9
ORIE3500/5500 Summer2011 Li
1
Review of Independence Random Vectors
Recall that
X
1
, . . . , X
n
are said to be
independent random variables
if
for any sets
A
1
, . . . , A
n
and each
A
j
⊂ ℜ
,
j
= 1
, . . . , n
:
P
[
X
1
∈
A
1
, . . . , X
n
∈
A
n
] =
P
[
X
1
∈
A
1
]
···
P
[
X
n
∈
A
n
]
.
In particular,
X
and
Y
are independent discrete random variables if
P
(
X
=
x, Y
=
y
) =
P
(
X
=
x
)
P
(
Y
=
y
);
X
and
Y
are independent continuous
random variables if
f
X,Y
(
x, y
) =
f
X
(
x
)
f
Y
(
y
).
Examples that use the deﬁnition of independence:
Example
A man and a woman decide to meet at a certain location. If each of them
independently arrives at a time uniformly distributed between 12 noon and
1pm. Find the probability that the ﬁrst to arrive has to wait longer than 10
minutes.
Solution: Let
X
and
Y
denote the time past 12 that the man and the woman
arrive. Then
X
and
Y
are independent random variables, each of which is
uniformly distributed over (0
,
60). Hence the answer is
2
P
(
X
+10
< Y
) = 2
∫ ∫
x
+10
<y
f
(
x, y
)
dxdy
= 2
∫
60
10
∫
y

10
0
(1
/
60)
2
dxdy
= 25
/
36
.
Examples to check whether two random variables are independent or not:
Example
If (
X, Y
) are jointly distributed with joint density
f
X,Y
(
x, y
) = 6
xy
2
,
0
< x, y <
1
,
then are
X
and
Y
independent?
To check this we would need to ﬁnd the marginals of
X
and
Y
ﬁrst. We
do that by integrating over the other variable.
f
X
(
x
) =
∫
1
0
6
xy
2
dy
= 2
x,
0
< x <
1
,
and
1
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f
Y
(
y
) =
∫
1
0
6
xy
2
dx
= 3
y
2
,
0
< y <
1
.
To see if the two random variables are independent or not we need to check
if the joint density is the product of the two marginals. Indeed we get
f
X,Y
(
x, y
) =
f
X
(
x
)
f
Y
(
y
)
,
and hence
X
and
Y
are independent.
Example
Let us consider a little variation of the previous example. If (
X, Y
) have joint
pdf
f
X,Y
(
x, y
) = 15
xy
2
,
0
< y < x <
1
.
Here although the part 15
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 Spring '11
 SONG
 Probability theory, Lily, 10 minutes, (yj )

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