F
Y
(
y
)
＝
F
(
+
,
y
)
＝
＝
P
{
Y
y
}
3.2 Marginal distribution
F
X
(
x
)
＝
F
(
x
,
+
)
＝
＝
P
{
X
x
}
Marginal distribution function for bivariate
Define –P57
t
h
e
m
a
r
g
i
n
a
l
c
d
f
s
o
f
(
X
,
Y
)
w
i
t
h
r
e
s
p
e
c
t
t
o
X
a
n
d
Y
r
e
s
p
e
c
t
i
v
e
l
y
Example 1. Suppose that the joint distributionof (X,Y) is specified by
Determine FX(x) and FY(y)。
Marginal distribution for discretedistribution
S
u
p
p
o
s
e
t
h
a
t
(
X
,
Y
)
～
P
{
X
＝
x
i
,
Y
＝
y
j
,
}
＝
p
i
j
，
i
,
j
＝
1
,
2
,
…
D
e
f
i
n
e
-
P
5
7
P
{
X
＝
x
i
}
＝
p
i
.
＝
，
i
＝
1
,
2
,
…
P
{
Y
＝
y
j
}
＝
p
.
j
＝
，
j
＝
1
,
2
,
…
the marginal pmf of (X, Y) with respect to X and Yrespectively.
Marginal density function
t
h
e
m
a
r
g
i
n
a
l
p
d
f
o
f
(
X
,
Y
)
w
i
t
h
r
e
s
p
e
c
t
t
o
X
a
n
d
Y
.
E
x
a
m
p
l
e
3
.
4
-
P
5
9
S
u
p
p
o
s
e
t
h
a
t
(
X
,
Y
)
～
f
(
x
,
y
)
,
(
x
,
y
)
R
2
,
d
e
f
i
n
e
the marginal pdf of Y
(
1
)
B
i
v
a
r
i
a
t
e
u
n
i
f
o
r
m
d
i
s
t
r
i
b
u
t
i
o
n
B
i
v
a
r
i
a
t
e
(
X
,
Y
)
i
s
s
a
i
d
t
o
f
o
l
l
o
w
u
n
i
f
o
r
m
d
i
s
t
r
i
b
u
t
i
o
n
i
f
t
h
e
d
e
n
s
i
t
y
f
u
n
c
t
i
o
n
i
s
s
p
e
c
i
f
i
e
d
b
y
By the definition, one can easily to find that if (X, Y) isuniformly distributed, then
Suppose that (X,Y) is uniformlydistributed on area D, which isindicated by the graph on the righthand, try to determine:
(1)the density function of (X,Y)；
(2)P{Y<2X} ；
(3)F(0.5,0.5)
Answer
where，1、2 are constants and 1>0,2>0、| |<1 are also constant , then, it is said that (X, Y)follows the two-dimensional normal distributionwith parameters 1, 2, 1, 2, and denoted it by
(2)Two dimensional normal distribution
Suppose that the density function of (X, Y) is specified by
Example
The joint pdf of (X,Y) is
Find the marginal pdf of X and Y.
Solution
temporary fix
when
when
so
temporary fix
temporary fix
temporary fix
when
when
So
Homework : P67: 5,6