UCLA
EE131A (KY)
1
EE 131A
Probability
Professor Kung Yao
Electrical Engineering Department
University of California, Los Angeles
Lecture 111
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UCLA
EE131A (KY)
2
Jointly pdf and marginal pdf (1)
•
Let F
XY
(x,y) be the joint cdf
of the joint rv’s
(X,Y),
where X and Y can be continuous or discrete rv’s.
The
joint pdf f
XY
(x,y)
of (X,Y) is given by
•
Marginal pdf f
X
(x)
of X is obtained from f
XY
(x,y) as
•
Marginal pdf f
Y
(y)
of Y is obtained from f
XY
(x,y) as
2
XY
XY
d
f
(x,y) =
F
(x,y), 
< x <
, 
< y <
.
dxdy
X
XY

f
(x) =
f
(x,y) dy , 
< x <
.
Y
XY

f
(y) =
f
(x,y) dx , 
< y <
.
UCLA
EE131A (KY)
3
Jointly pdf and marginal pdf (2)
Ex. 1. (Ex. 2, p. 10, Lec102).
Then
XY
1,
0
x
1, 0
1,
f
(x,y) =
0,
elsewhere .
y
1
X
XY
X

0
1
Y
XY
Y

0
2
XY
XY
d
f
(x) =
f
(x,y) dy =
1 dy = 1 =
F (x),
0
x < 1.
dx
d
f
(y) =
f
(x,y) dx =
1 dx = 1 =
F (y),
0
y < 1.
dy
1, 0
x < 1, 0
y < 1,
d
f
(x,y) =
F
(x,y) =
0,
elsesher
dxdy
e.
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