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AMS 553: Homework 1
X
be a continuous random variable with pdf
f
(
x
) =
x
2
+
2
3
x
+
1
3
for 0
≤
x
≤
c
(a) Find the value of
c
; (b) Plot the pdf
f
(
x
); (c) Compute and plot the cdf
F
(
x
); (d) Compute
P
(
1
3
≤
X
≤
2
3
),
E
[
X
], and
V ar
(
X
).
X
and
Y
are jointly discrete random variables with
p
(
x,y
) =
‰
2
n
(
n
+1)
for
x
= 1
,
2
,...,n
and
y
= 1
,
2
,...,x
,
0
otherwise.
Compute the pmfs
p
X
(
x
) and
p
Y
(
y
) and determine whether
X
and
Y
are independent.
X
and
Y
are jointly continuous random variables with density function
f
(
x,y
) =
‰
32
x
3
y
7
if 0
≤
x
≤
1 and 0
≤
y
≤
1,
0
otherwise.
Compute
f
X
(
x
) and
f
Y
(
y
) and determine whether
X
and
Y
are independent.
X
is a discrete random variable with
p
X
(
x
) = 0
.
25 for
x
=

2
,

1
,
1
,
2. Let
Y
also
be a discrete random variable such that
Y
=
X
2
. Show that
Cov
(
X,Y
) = 0. Therefore, uncorrelated
random variables are not necessarily independent.
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This note was uploaded on 01/31/2011 for the course AMS 553 taught by Professor Badr,h during the Spring '08 term at SUNY Stony Brook.
 Spring '08
 Badr,H

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