Jacobs University Bremen
School of Engineering and Science
Peter Oswald
Fall Term 2010
120201 ESM3A — Problem Set 9
Issued: 10.11.2010
Due: Wednesday 17.11.2010 (in class)
This problem set is about several random variables (joint distributions, independence, con
ditioning).
9.1.
Random variables
X
and
Y
are said to be jointly continuous if there is a function
f
X,Y
:
IR
2
→
[0
,
∞
), called the
joint density
, such that
P
(
X
∈
(
a,b
Y
∈
(
c,d
]) =
Z
d
c
Z
b
a
f
X,Y
(
x,y
)d
x
d
y
for every
a
≤
b
and
c
≤
d
.
(a) Show that jointly continuous random variables are independent if and only if we have
f
X,Y
(
x,y
) =
f
X
(
x
)
f
Y
(
y
) for all such
x,y
∈
IR, where these functions are continuous.
(b) Give an example of two jointly continuous random variables that are
not
independent.
9.2.
Let the joint probability density function of
X
and
Y
be given by
f
X,Y
(
t,s
) = 12
ts
2
,
0
≤
s
≤
t
≤
1
,
f
X,Y
(
t,s
) = 0
otherwise
.
(a) Determine if
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This note was uploaded on 02/06/2011 for the course ESM 3A taught by Professor Prof. dr. peter oswald during the Spring '11 term at Jacobs University Bremen.
 Spring '11
 Prof. Dr. Peter Oswald

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