Consider the pair of random variables ( X , Y ) with joint PDF f X , Y ( x , y ) = 2 for 0 ≤ x ≤ y ≤ 1 and f X , Y ( x , y ) = 0 otherwise. (In particular, note that the joint PDF is only nonzero when x ≤ y .) (a) Draw a picture of the range of ( X , Y ) in the x y -plane. (b) Describe the conditional distribution of X given Y = y , for 0 ≤ y ≤ 1 . (c) What is the marginal PDF f X of X ? (d) Are X and Y independent?
Consider the pair of random variables ( X , Y ) with joint PDF f X , Y ( x , y ) = 2 for 0 ≤ x ≤ y ≤ 1 and f X , Y ( x , y ) = 0 otherwise. (In particular, note that the joint PDF is only nonzero when x ≤ y .) (a) Draw a picture of the range of ( X , Y ) in the x y -plane. (b) Describe the conditional distribution of X given Y = y , for 0 ≤ y ≤ 1 . (c) What is the marginal PDF f X of X ? (d) Are X and Y independent?
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.2: Expected Value And Variance Of Continuous Random Variables
Problem 10E
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Question
Consider the pair of random variables
(
X
,
Y
)
with joint PDF
f
X
,
Y
(
x
,
y
)
=
2
for
0
≤
x
≤
y
≤
1
and
f
X
,
Y
(
x
,
y
)
=
0
otherwise. (In particular, note that the joint PDF is only nonzero when
x
≤
y
.)
(a) Draw a picture of the range of
(
X
,
Y
)
in the
x
y
-plane.
(b) Describe the conditional distribution of
X
given
Y
=
y
, for
0
≤
y
≤
1
.
(c) What is the marginal PDF
f
X
of
X
?
(d) Are
X
and
Y
independent?
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